The emergence of two-dimensional transition metal dichalcogenides (2D-TMDs) offers exciting opportunities for probing new physical phenomena and potential applications in catalysis,^[1,2] optoelectronics,^[3,4] nanoelectronics,^[5–7] etc. For example, Maeso et al.^[4] recently investigated the electrical and optical properties of vertical few-layer MoS₂ and found that the light absorption can be improved by fabricating vertical photodevices using few-layer flakes, achieving a photoresponse up to 0.11 A⋅W⁻¹ and an external quantum efficiency up to 30%. Shahraki et al.^[6] reported that MoS₂ multilayers have unique mechanical and electrical properties, which can be a promising channel material for the n-type piezoelectric field-effect transistor. However, considering that the devices constructed of TMDs are often worked under various environments,^[8–12] it is necessary to explore the change of related material properties under the external stimulus.

In particular, external pressure provides a viable avenue to modulate the crystal structure and electronic properties in a clean and precisely controllable manner. Currently, a series of experiments^[13–19] and theories^[20–22] showed that the phase transition of 2H_c-to-2H_a and metallization in MoS₂ can be induced by hydrostatic pressure. Especially, the exciting properties of pressure-induced superconductivity emerged in 2H_a-MoS₂ and proved that 2H_a-MoS₂ is a high-pressure stable phase,^[23] while the phase transition of 2H_c-to-2H_a is absent in MoSe₂, MoTe₂, and WS₂ under the condition of hydrostatic pressure.^[22,24–27] Therefore, it is an urgent issue to explore the pressure-dependent phase transition of MoS₂ from 2H_c-to-2H_a. For the case of bulk MoS₂, many experimental measurements showed the transition of 2H_c-to-2H_a, associating with a semiconducting-to-metallic state, but there are large differences in the values of transition pressure measured by different experimental groups (18.5–26 GPa).^[13–19] Also, the structural transition of 2H_c-to-2H_a and metallization in bulk MoS₂ under pressure have been reported by using first-principle calculations and molecular dynamics simulations and the values of pressure are in the range of 13 GPa to 20 GPa.^[13,20–22] In addition, the transition pressure of MoS₂ from 2H_c-to-2H_a under hydrostatic pressure increases from 19.0 GPa to 36.0 GPa as the thickness reduces from bulk to bilayer.^[19] Meanwhile, MoS₂ undergoes a metallization transition upon hydrostatic pressure, while the transition pressure decreases as the number of layers increases.^[28,29] However, there are great differences (13–26 GPa) in transition pressure between some experiments and theories for MoS₂. Furthermore, the potential mechanisms of structural transition, pressure metallization, and related evolution in MoS₂ have not been comprehensively understood. In particular, the pressure-size phase diagram is still unclear.

In this paper, we investigate the structural transition and metallization of MoS₂ under pressure based on the bond relaxation method and thermodynamics. It is found that the transition pressures of 2H_c-to-2H_a and metallization increase significantly with a decreasing number of layers, while the change rate of structural transition is greater than that of metallization. Moreover, we establish a size-pressure phase diagram of MoS₂, identifying the transition mechanism of the structural properties.

MoS₂ consists of intralayer strong covalent bonding and is stacked with interlayer by weak vdW bonding. Noticeably, the structural transition of 2H_c-to-2H_a originates from the interlayer sliding. Conventionally, the Gibbs free energy G_sum (P) of MoS₂ under hydrostatic pressure is G_sum (P) = G₀ + G(P), where G₀ is a reference value at ambient pressure, and G(P) is a part of the pressure-induced energy storage of unit cell. Since G(P) = – ∫ SdT + ∫ V dP and the temperature is almost unchanged under the approach of pressure, the equation can be simplified into G(P) = ∫ V d P.

Under the condition of hydrostatic pressure, the lattice strain can be expressed as ε (x,y,z) = l/l₀ – 1, where l and l₀ denote the length after and before relaxation along x-, y-, and z-directions. For the intralayer of 2H_c and 2H_a phases, it is feasible to take the elastic modulus as a constant in a small strain range due to the strong ionic-covalent bonds, while assuming that sliding of interlayer does not change the properties with intralayer.^[30,31] Therefore, the pressure-dependent strain of intralayer along with x-, y-, and z-directions can be shown as

In general, the interlayer interactions of the adjacent layers in MoS₂ include three kinds of modes, i.e., S–S, S–Mo, and Mo–Mo, and the interaction can be described by the Lenard-Jones potential, i.e., U(r) = –Γ [ (σ/r_ij)¹² – (σ/r_ij)⁶ ], where r_ij denotes the distance between i and j atoms. (σ/r_ij)¹² and (σ/r_ij)⁶ are the pairwise repulsion and attraction, respectively. For the nearest-neighbor interaction, the equilibrium spacing r₀ is related to σ as .^[32,35] Γ corresponds to the depth of the energy well at the equilibrium separation.^[32,36] In our case, we take Γ(σ) of S–S, S–Mo, and Mo–Mo as 0.024 eV (3.13 Å), 0.0028 eV (3.67 Å), and 0.00059 eV (4.20 Å), respectively.^[32,35,36] The total van der Waals (vdW) interaction energy can be written as . Moreover, the average interlayer interaction of unit cell for two phases of 2H_c and 2H_a through taking into account of the nearest neighbor atoms is

Generally, the thermal stability of a system is determined by the cohesive energy.^[37,38] Moreover, the transition pressure (P_c ) of the specimen correlates to the cohesive energy, i.e., , where z_i and E_i are the coordination number (CN) and the single bond energy for all atoms, respectively. D is the thickness of MoS₂. Clearly, the compression of the bond length and the bond strength will be stronger under the condition of pressure, resulting in the increasment of the transition pressure, while decreasing thickness leads to the reduction of the transition pressure because of the lower cohesive energy.^[38,40] Therefore, the competition between energy enhancement induced by external pressure and reduction driven by the thickness down to nanoscale determines the transition pressure.

In the case of pressure, the entire atomic bonds of the specimen become shorter and stronger because of volume shrinkage and deformation energy storage. Therefore, according to the integral of the volume-pressure profile, the pressure-induced energy storage is

In addition, in light of the atomic-bond-relaxation (ABR) method,^[41,42] the atoms located at the edge will relax spontaneously. Therefore, the intra-atomic potential well depresses from E_b to , where E_b denotes the single bond energy of the bulk counterpart, and the index m is the bond nature factor. c_i = 2/( 1 + exp (( 12 – z_i)/8z_i)) is the bond contraction coefficient.^[41,42] Thus, the thickness-dependent transition pressure is

On the other hand, the variation of the bond parameters in MoS₂ would induce the change of the system energy, and the interaction potential of intralayer consists of bond-stretching energy U_bond, bond angle variation energy U_angle, and Coulomb electrostatic energy U_coul,^[43,44]

Combining the interaction potentials of intralayer and interlayer, the pressure-induced variation of the average single bond energy in MoS₂ can be shown as , where ΔU_tra is the single bond energy variation of the intralayer, and is the interlayer interaction of an S atom. The superscripts 1 and 2 of the interlayer energy represent the 2H_c and 2H_a phases, respectively. Thus, the cohesive energy of the system under the pressure is

For the case of 2H_c-MoS₂, the Mo atoms in one sheet are on the top of the S atoms in the next, while there is a misalignment between S atoms of the adjacent layer for 2H_a-MoS₂(Fig. 1(a)). It is found that increasing hydrostatic pressure leads to a decrease of the lattice parameter a (the bond length d) and an increase of the thickness of the intralayer (Fig. 1(b)). The variation trends of bond angle in the opposite directions are the consequence of changes from in-plane θ and out-of-plane ψ with increasing pressure (Fig. 1(c)). The changes of lattice parameter and bond length are approximately linear due to strong covalent bonding between the Mo and S atoms, while an expansion of intralayer with the pressure is due to the effect of negative Poisson’s ratio. In light of the ABR consideration,^[41,42] the interlayer distances are 2.99 Å and 3.05 Å for 2H_c and 2H_a phases at ambient pressure, respectively (the inset of Fig. 1(d)). In detail, the phase transition does not change the interaction of S–S, while the interaction distances of Mo–S and Mo–Mo have been changed. Therefore, the interlayer distance is determined by the interactions of Mo–S and Mo–Mo. However, the interactions of Mo–S and Mo–Mo are weak at the interlayer, resulting in a small difference between the two phases. Notably, the interlayer distance of MoS₂ decreases significantly and non-linearly with increasing pressure, reflecting that the interlayer is associated with weak vdW bonding (Fig. 1(d)). Importantly, the change rate decreases with reducing interlayer distance due to the strong vdW interaction caused by the reduction of the interlayer distance. Furthermore, the variation range of 2H_a-MoS₂ is slightly larger than that of 2H_c-MoS₂, suggesting that the stacking order affects the interlayer interaction. Evidently, the changes in bond parameters are consistent with the available results from experimental measurements and theoretical calculations.^[21,45,46]

Fig. 1.

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	Fig. 1. (a) The schematic diagrams of atomic arrangement of 2Hc- and 2Ha-MoS2 structures, where the Mo and S atoms are represented by blue and yellow balls, respectively. (b) Pressure-dependent variations of lattice parameter a, intralayer S–S distance c, and Mo–S bond length d. (c) Bond angles θ and ψ as a function of pressure. (d) Interlayer distance t as a function of pressure.

Figure 2(a) shows the relationship between pressure and unit volume of bulk MoS₂ in 2H_c and 2H_a phases. It is observed that the volume decreases gradually with increasing pressure and the volume of 2H_c is larger than that of 2H_a under high pressure as the variation of the volume is mainly contributed by the interlayer separation. In addition, fitting the curves of P–V by the third-order Birch–Murnaghan equation of state ,^[47,48] we obtain the bulk moduli of 2H_c and 2H_a phases as B₀ = 76.06 ± 0.11 GPa (B’ = 3.12 GPa) and B₀ = 85.13 ± 0.21 GPa (B’ = 2.57 GPa), respectively. The bulk moduli obtained in our studies are qualitatively in agreement with the results reported by Aksoy et al.^[15] and Bandaru et al.^[16] Figure 2(b) shows the pressure-dependent difference of Gibbs free energy of 2H_a- to 2H_c-MoS₂. The Gibbs free energies of the two phases cross near 20.6 GPa, indicating that the structural transition will take place. Notably, the large difference of transition pressure in some experiments may be due to two main factors: (1) different types of defects may appear in the samples, such as vacancies or dislocations, which affect the interlayer interaction; (2) different characterized methods, such as Raman spectra and x-ray diffraction, may lead to the differences.^[14,17–19] In nature, reducing the size to nanoscale, the surface state of a nanomaterial is different from that of the bulk counterpart due to the low coordination numbers of the surface atoms, which infers that the transition pressure will be significantly affected by the surface effect. Figure 2(c) shows the layer-dependent transition pressure of MoS₂ from 2H_c-to-2H_a. Clearly, the transition pressure is significantly enhanced from 20.6 GPa to 34.8 GPa as the thickness reduces from bulk to bilayer. Originally, the increase of the transition pressure can be attributed to the competition between pressure-induced enhancement and thickness-induced reduction of atomic cohesive energy. Similarly, Cheng et al.^[19] demonstrated that the transition pressure of 2H_c-to-2H_a in MoS₂ can be continuously tuned from 19 GPa to 36 GPa by reducing the thickness from bulk down to bilayer by Raman analysis. As shown in the inset of Fig. 2(c), the phase transition from 2H_c-to-2H_a is through the sliding of the interlayer, which changes the stacking order of the adjacent layer. Generally, this structural transition is absent in the case of monolayer duo to lack of interlayer interaction.

Fig. 2.

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	Fig. 2. (a) Pressure-dependent unit volume of MoS2 in both 2Hc and 2Ha phases. (b) Pressure-dependent relative Gibbs free energy of 2Ha- and 2Hc-MoS2 phases. (c) Phase diagram of MoS2 from 2Hc-to-2Ha as a function of thickness. The solid line is the dividing line.

Figure 3 depicts the pressure-dependent energy gain and band shift in MoS₂. Especially, the bond stretching and bond angle relaxation are monotonically decreasing, and the electrostatic interaction increases approximately linearly with increasing pressure, as shown in Fig. 3(a). It is obvious that the bond stretching energy and bond angle relaxation energy are negative, while the electrostatic interaction energy is positive. In addition, the interlayer interaction of S atom decreases approximately linearly with increasing pressure (Fig. 3(b)). Notably, the change rate of the interlayer interaction energy of 2H_a with increasing pressure is greater than that of 2H_c. In general, when the system is perturbed by the external environment (such as doping and pressure), the system will relax to a new self-equilibrium state. In our case, the bandgap of bulk MoS₂ is red-shift with increasing pressure due to strong interlayer interactions and metallization at pressures of 23 GPa and 21 GPa for the 2H_c and 2H_a phases, respectively.

Fig. 3.

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Fig. 3. (a) The pressure-dependent energy gain of single Mo–S bond including bond stretching, bond angle distortion, and electrostatic interaction energies. (b) The pressure-dependent interlayer interaction energy of an S atom in 2Hc and 2Ha phases. (c) The metallization transition of MoS2 as a function of layer number. The solid line is the dividing line. The inset is the pressure-dependent bandgaps for monolayer, bilayer, bulk 2Hc- and 2Ha-MoS2.

For the cases of monolayer and bilayer MoS₂, the bandgap blue-shifts first and then red-shifts with increasing pressure as shown in the inset of Fig. 3(c). In fact, the physical origin can be ascribed to the limited interlayer interaction and positive electrostatic interaction energy. Interestingly, the metallization of monolayer 2H-MoS₂ under pressure emerges at a pressure of 65.0 GPa, while for the bilayer MoS₂ with 2H_c and 2H_a phases, the metallization will be observed at 38.7 GPa and 37.9 GPa, respectively. Importantly, it can be clearly seen that the transition pressure of semiconductor-to-metal decreases as the number of layers increases and the metallization of the 2H_a phase is easier to achieve than that of the 2H_c phase (Fig. 3(c)). Similarly, Nayak et al.^[28] and Kim et al.^[29] found that the transition pressure decreases as the number of layers increases based on the density functional theory. Essentially, for 2H-MoS₂, the valence band maximum (VBM) and the conduction band minimum (CBM) are mainly dominated by the d orbitals of the Mo atoms and p orbitals of the S atoms. Under the hydrostatic pressure, the strong hybridization between the d orbitals (mainly d_z², d_xy, and d_{x² – y²} ) of Mo atoms and p orbitals of S atoms will result in the overlap between VBM and CBM as well as metallization.^[28,29] Moreover, the stacking order between layers has a direct relationship with the transition pressure for MoS₂. Therefore, the system undergoes a series of phase transitions under pressure, which is from the semiconducting 2H_c-to-2H_a phases and to the metallic 2H_a phase with the pressure up to 70 GPa.

Based on the discussion mentioned above, we further establish a size-pressure phase diagram of 2H-MoS₂, as shown in Fig. 4. Also, we obtain the values of two types of phase transitions with different cases, which are in good agreement with the previous results (Table 1). Evidently, the sliding of interlayer takes place at the structural transition from 2H_c-to-2H_a and the transition pressure increases with a decreasing number of layers. In addition, the metallization arises from the overlap of CBM and VBM owing to the variation of the crystal potential induced by the change of bond identities. For multilayer and bulk MoS₂, the interlayer interaction is found to control the electronic structure, while for the case of monolayer MoS₂, the changes in the electronic structure under pressure are mainly controlled by the bond length and bond angle. Notably, for n > 10 layers, the structural transition and metallization in MoS₂ are almost complete simultaneously and the transition pressures are 20.6 GPa and 21 GPa, respectively. However, for n < 10 layers, the material becomes metallic when the structural transition is complete.

Fig. 4.

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	Fig. 4. Phase diagram of MoS2 under the condition of hydrostatic pressure. The blue line represents the boundary between 2Hc and 2Ha phases. There is no interlayer interaction in monolayer represented by 2H phase. The semiconductor to metal transition is separated by the red line, while the pale green region denotes semiconducting.

Table 1.

Table 1.

Table 1. Calculated transition pressure (in GPa) of two types of phase transitions of MoS2 for different cases. . Phase transition types Monolayer Bilayer Trilayer Bulk 2Hc-to-2Ha our results – 34.8 30.6 20.6 previous results – 36.0d – 19.0a, 23.0b, 26.6b, 20.5c Semiconducting-to-metallic our results 65.0 37.9 33.4 21.0 previous results 67.9e 39.2e 29.5e 19.0a, 22.3e a Ref. [13], b Ref. [14], c Ref. [15], d Ref. [19], e Ref. [28].

Table 1. Calculated transition pressure (in GPa) of two types of phase transitions of MoS2 for different cases. .

In summary, we establish a theoretical model to investigate the structural properties of MoS₂ by bond relaxation and thermodynamic considerations. We find that the structural transition of bulk MoS₂ from 2H_c-to-2H_a is determined by the interlayer coupling, while for the case of few-layers, the surface effect plays the dominant role in the transition pressure. The competition between energy enhancement induced by external pressure and reduction motivated by the thickness down to the nanoscale results in the significant transition pressure increase with decreasing thickness. Our predictions agree well with the available evidence, which suggests perspectives in a deep understanding and controlling structure design in two-dimensional materials.