Two-dimensional (2D) material is a rapidly expanding research area. The great success of graphene has ignited intensive interest in other 2D materials. The layered transition metal dichalcogenides (TMDs) is one of the most widely researched classes of 2D materials beyond graphene. The TMDs are normally represented with the formula MX₂, where X is a chalcogen atom (S, Se, or Te), and M is a transition metal atom from groups IV–X (such as Ti, Mo, and W), as shown in Fig. 1.^[1] In unit cell of TMDs, a transition metal atom M is sandwiched between two chalcogen atoms X, as shown in Fig. 2.

Fig. 1.

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	Fig. 1. (color online) Transition metals and the three chalcogen elements in the TMD layered structures. Reprinted with permission from Ref. [1], copyright (2013) by Macmillan Publishers Limited.

Fig. 2.

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	Fig. 2. (color online) (a) Structure of monolayer MoS2. Side view (b) and top view (c) of monolayer MoS2. The horizontal orientation is armchair direction, the vertical orientation is zigzag direction. The atoms in purple and yellow color are Mo and S, respectively. Reprinted with permission from Ref. [2], copyright (2013) by the Frontiers of Physics.

TMDs have gained significant interest due to their unique optical, electronic, mechanical and thermal properties.^[3] The monolayers of all the members in this family are semiconducting with bandgaps of around 1–2eV,^[4] which makes them prime candidates for electronic devices. Furthermore, unlike the indirect bandgaps for bulk TMDs, the monolayer TMDs has direct bandgaps due to their ultrathin structures (quantum confinement), which induce the changes of electronic structure in monolayer TMDs. The direct bandgap indicates that they are also well suited for optoelectronics. Indeed a wide range of different TMD-based optoelectronic devices have already been successfully fabricated, such as high-performance field-effect and thin film transistors,^[5] photovoltaics,^[6] and photodetectors.^[7] As a member of 2D TMD family, monolayer MoS₂, has a large direct bandgap of 1.8 eV. For MoS₂ transistor, through various processing techniques, such as improving sample quality, removing absorbates, or depositing atop a high-dielectric layer, its room temperature carrier mobility can be significantly enhanced,^[8–11] and close to its theoretically predicted phonon-limited intrinsic value (about ).^[12,13] Very recently, the growth of MoS₂ on insulating substrates was also realized, which enables atomically thin high-performance optoelectronic and electronic devices can be fabricated without film transfer.^[14]

Thermal management is an important issue for the design and application of electronic devices, which even is becoming a limiting factor currently to the device development.^[15–29] Owing to the atomic thickness of monolayer TMDs, thermal management is more challenging in TMDs-based electronics. On one hand, highly localized Joule heating in their ultrathin confined space can easily create “hot spots”.^[30,31] On the other hand, previous experimental^[32–34] and theoretical studies^[2,35–40] revealed that the thermal conductivities of monolayer TMDs are 2–3 orders lower than that of graphene. These two limitations cause a crucial bottleneck for efficient thermal management of TMD-based devices. Furthermore, similar to electron scattering at an interface, phonon scattering at an interface also plays a critical role in the nanoscale devices, which is even more crucial than the material itself.^[41–43] Thus, low thermal conductivity in monolayer TMDs and thermal conductance at the interface will significantly affect performance and reliability of the TMDs-based devices.

In this article, we review the recent advances in the study of the dynamic and thermal properties of 2D TMDs from both experimental and theoretical points of view. The rest of this article is organized as follows: section 2 introduces the phonon dispersion of TMDs, section 3 discusses phonon relaxation time and mean free path of TMDs, section 4 and 5 are devoted to the discussion of thermal conductivity and interfacial thermal conductance of TMDs, respectively. In section 5, we present a brief conclusion.

Since phonon energy is essentially the energy of atomic vibrations, to understand the thermal properties of TMDs, it is necessary to firstly inspect the lattice vibrational modes (phonons) of the materials. MoS₂ is a typical layered dichalcogenide, and the most studied material in this family. Each MoS₂ unit cell contains 3 (1 Mo and 2 S) atoms, which leads to the formation of 3 acoustic (A) and 6 optical (O) phonon modes as shown in Fig. 3(a).^[44] The three acoustic phonon modes are longitudinal acoustic (LA) mode, transverse acoustic (TA) mode, and flexural acoustic (ZA) mode. LA mode corresponds to compressional waves, in which atomic displacements are along the wave propagation direction; TA mode corresponds to shear waves, in which in-plane displacements are perpendicular to the propagation direction, and ZA mode corresponds to out-of-plane atomic displacements. The phonon dispersion is the relationship between the phonon wave vector and phonon energy E or frequency ω ( , where is the reduced Planck constant).

Fig. 3.

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	Fig. 3. (color online) Phonon dispersion and projected PDOS for (a) MoS2, (b) MoSe2, and (c) WS2. Reprinted with permission from Ref. [44], copyright (2016) by the Royal Society of Chemistry.

The phonon dispersion and density of states (DOS) of MoSe₂ and WS₂ are also shown in the Figs. 3(b) and 3(c). The low-frequency acoustic phonon branches of MoS₂ up to 233 cm⁻¹ are mainly from Mo(xy), S(xy), and Mo(z) vibrations due to similar mass between Mo and 2S, whereas those of WS₂ up to 182.3 cm⁻¹ are mainly from the W(xy) and W(z) vibrations due to the much larger mass of W atoms.^[44] These results are similar to the diatomic linear chain model, that the scale of the acoustic phonon branch/vibration is dominated by atoms with larger mass; however, the optical phonon branch/vibration is dominated by the atoms with smaller mass.^[44]

At the low wave vector near the center of Brillouin zone, the frequency of the LA and TA modes have linear dispersions^[45,46] as and , respectively. The group velocity and m/s are about five times lower than those (5953 m/s for LA, and 3743 m/s for TA) in graphene.^[36] However, the ZA modes have an approximately quadratic dispersion,^[45–47] The phonon group velocity with relationships of and ω along the −M high symmetrical line are shown in Fig. 4. For the LA and TA modes, the group velocities dramatically decrease with the increase of phonon frequency; while for the ZA mode, its group velocity increases with increasing frequency, and reaches the maximum value at the middle point of the –M line, then decreases to zero at the zone edge.^[36]

Fig. 4.

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	Fig. 4. (color online) Phonon group velocity of acoustic phonon modes along the –M direction in MoS2 with a function of wave number (a) and frequency (b). Reprinted with permission from Ref. [36], copyright (2014) by American Physical Society.

Phonons have relaxation times as they travel through the materials due to various scattering processes, such as phonon–phonon umklapp scattering, boundary scattering, and defects scattering. Since extrinsic boundary scattering and defects scatting can be removed by improving material quality, the phonon–phonon umklapp scattering is highly associated with the intrinsic phonon relaxation process/time, which can be derived through Klemensʼs time-dependent perturbation theory^[48,49]

Fig. 5.

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	Fig. 5. (color online) (a) Relaxation time and (b) mean free path for LA, TA, and ZA modes in monolayer MoS2 with a function of frequency. Reprinted with permission from Ref. [36], copyright (2014) by American Physical Society.

Phonon mean free path (MFP) is a key quantity for understanding the size-dependent thermal properties. The umklapp scattering limited MFP λ can be obtained, based on the mode relaxation time τ and group velocity v. The phonon MFP for the mode at the point with s polarization is defined as . Figure 5(b) shows the phonon MoS₂ as a function of frequency and phonon mode. The MFP decreases obviously with the increase of frequency. In the entire frequency regime, LA mode has the longest MFP among the LA, TA, and ZA modes, due to the combined effects of the frequency-dependent relaxation time and group velocities. The largest dominated MFP among the zone-center acoustic modes is LA mode with a value of around 18.1 nm, which is much large than that of TA mode, around 5.0 nm, which is consistent with the value of 5.2 nm by the molecular dynamics (MD) calculations.^[2,36]

The thermal conductivity κ of monolayer TMDs can be obtained based on Fourierʼs law, , where is the heat flux per unit area and is the temperature gradient along the heat flux direction. The thermal conductivity can also be related to the specific heat by , where is specific heat, v and λ are the average phonon group velocity and MFP, respectively.^[50] This expression is commonly used under diffusive transport conditions, where material sizes are much larger than the phonon MFP ( ).^[47] During the calculation of heat flux, normally the thickness of 2D materials h is chosen to be the inter-layer spacing in the corresponding bulk materials (e.g., h = 4.41 Å for MoS₂^[12]).

The low thermal conductivities of TMDs have been reported in both theoretical^[2,36,44] and experimental^[32–34,51] studies. Table 1 summarizes the reported thermal conductivities of monolayer TMDs along with their methodologies of measurement. At room temperature, the superior theoretical thermal conductivity of monolayer MoS₂ is about obtained by calculating Boltzmann transport equation (BTE).^[52] Its thermal conductivity is about , obtained by solving the non-equilibrium Greenʼs function (NEGF).^[37] Its experimental result is .^[33] It can be found that there is a large variation between the results, which is mainly attributed to different approaches applied during the measurement. Each approach has its own advantages and disadvantages. For experimental approaches, the thermal conductivity of sample is sensitive to the defects or the substrates, which introduce extra phonon scattering and decrease the intrinsic thermal conductivity. In NEGF approach, only the harmonic force constants have been used to consider ballistic or semi-ballistic phonon transport, which ignores the anharmonic phonon–phonon scatting in the TMDs. BTE method is limited to first-order anharmonicity during the calculation. Molecular dynamics (MD) simulation is also widely used for the calculation of thermal conductivity, which naturally accounts for the lattice anharmonicity to all orders. However, the accuracy of MD simulation is limited by the quality of empirical interatomic potentials. For monolayer WSe₂, the thermal conductivity, about ,^[53] is one order of magnitude lower than that of MoS₂, which is due to the ultralow Debye frequency and heavy atoms mass in WSe₂. Similar to graphene, it has been found that the thermal conductivity of TMDs is also temperature dependent, which decreases with the increasing temperature.^[53,54]

Table 1.

Table 1.

Table 1. Summary of the thermal conductivities of different monolayer TMDs at room temperature. . TMDs Methodologies Type κ ( ) Ref. MoS2 exfoliated suspended 34.5 [33] MoS2 first principles calculation suspended 23.2 [36] MoS2 optothermal Raman technique suspended 84 [55] MoS2 Boltzmann transport equation supported 103 [52] MoS2 optothermal Raman technique supported on gold 55 [55] WS2 Boltzmann transport equation suspended 142 [52] WS2 Raman spectroscopy suspended 32 56 WS2 first principles calculation suspended 31.8 [44] MoSe2 Boltzmann transport equation suspended 54 [52] MoSe2 first principles calculation suspended 17.6 [44] MoSe2 optothermal Raman technique suspended 59 [55] MoSe2 optothermal Raman technique supported on SiO2 24 [55] WSe2 Boltzmann transport equation suspended 3.935 [53]

Table 1. Summary of the thermal conductivities of different monolayer TMDs at room temperature. .

Table 2.

Table 2.

Table 2. Summary of the interfacial thermal conductances between different monolayer MoS2 and other materials at room temperature. . TMDs Contacting materials Contact type Methodology ITC/ Ref. MoS2 Si side experimental 0.974 69 MoS2 SiO2 side experimental 1.7 65 MoS2 gold Au side theoretical 17.2 66 MoS2 gold Au edge theoretical 221 61 MoS2 h-BN side experimental 17.0 68 MoS2 graphene side theoretical 13.8 64 MoS2 graphene edge theoretical 6420 63

Table 2. Summary of the interfacial thermal conductances between different monolayer MoS2 and other materials at room temperature. .

It has been well known that the in-plane thermal conductivity of graphene decreases significantly when it is in contact with a substrate.^[47] Similar trends has also been reported for monolayer TMDs due to damping of the flexural acoustic phonon.^[55] Using optothermal Raman technique, the measured thermal conductivities of suspended monolayer MoS₂ and MoSe₂ are 84 and , respectively, at room temperature. However, when they are supported on SiO₂ substrate, their thermal conductivities decrease to 55 and , respectively.^[55]

Monolayer TMDs is promising in electronic and photonic device applications, such as transistors, light-emitters, and photodetectors.^[57] With the successful growth of MoS₂ on insulating substrates,^[14] and the significant improvement in the mobility,^[58–60] it is reasonably expected that high-performance field-effect transistors based on monolayer MoS₂ will be realized in the near future. Unlike the single transistor, owing to the atomic thickness of 2D materials, thermal management of their integrated devices is becoming vitally important. The Joule heating in the confined ultrathin 2D TMDs can easily create localized hot spots. Unfortunately, previous studies^[2,32–35] revealed that the monolayer TMDs have remarkable low thermal conductivity. Furthermore, phonon scattering at an interface also plays a critical role, which hinders the heat transport across the interface.^{[41–43,61–63]} combination of these limitations poses a significant challenge for efficient thermal management of TMDs-based devices, and interfacial thermal conductance (ITC) between TMDs and other (electrode) materials will adversely affect device reliability and performance.

In general, there are two types of interface geometry between TMD sheets and other material surface: side contact and edge contact.^[57]

5.2. Edge contact by covalent bonds

As discussed before, monolayer TMDs have very low in-plane thermal conductivities and interfacial thermal conductance at the side-contacted interface, which hinder the heat dissipation from TMDs-based integrated devices. A probable solution is to construct an edge-contacted interface with covalent bonds by fully utilizing the contacted relative high thermal conducting materials to speed up the heat dissipation of “hot-spot” in TMDs-based devices.

As a typical example, MoS₂ field effect transistor (FET), it has been reported that there exists a large contact resistance at the interface between the MoS₂ sheet and metal electrode with the side contact due to the van der Waals gap, which drastically restrains the drain current.^[57] However, for the edge contact, strong overlapping of electronic orbitals exists between the MoS₂ and metal electrode, which leads to a remarkable reduction in tunnel barrier and a significant increase in electrical current. The ITC has been significantly increased from (with side contact) to (with edge contact).^[61,66] So far both previous experimental^[70] and theoretical^[71] studies have also confirmed that the edge contact may play a key role in the efficient injection of electrons from metal to TMDs.

Although the thermal conductivity of monolayer MoS₂ is 2–3 orders lower than that of graphene, it is interesting to find that the covalent bonds formed at the MoS₂/metal interface enables its ITC to be comparable with that of graphene-metal interface. As shown in Fig. 6, different Au surfaces form different bonding configurations, causing large ITC variations, which is due to the different coordination numbers and structural symmetries at the interface. For such edge-contacted MoS₂/metal interface, it is reported that at room temperature, the ITC is primarily dominated by phonon transport, and the effect of electron-phonon coupling is negligible.^[61]

Fig. 6.

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	Fig. 6. (color online) Dependence of interfacial thermal conductance on Au crystal surfaces and contact directions. Reprinted with permission from Ref. [61], copyright (2016) by the Springer.

Furthermore, the ITC is found temperature-dependent. Figure 7 shows the results of the temperature-dependent ITC of MoS₂/Au interface. It is seen that the ITC and heat flux across the interface increases linearly with increasing temperature. However, the interfacial temperature jump is nearly independent of temperature. The trend of temperature dependence of ITC is possibly caused by the factor that, when the temperature is increased, more high-frequency phonons are excited, which contribute additional energy carriers for interfacial heat transport. Furthermore, the increasing temperature will increase the anharmonicity of atomic interactions at the interface. Eventually, the phonon transmission is enhanced through the inelastic phonon scattering. It is noted that a similar trend for such temperature-dependence was also observed in other interfaces.^[72]

Fig. 7.

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	Fig. 7. (color online) (a) Interfacial thermal conductance of MoS2/Au interface as a function of temperature; (b) interfacial temperature jump and heat flux as a function of temperature. Reprinted with permission from Ref. [61], copyright (2016) by the Springer.

Since phonon energy is essentially the energy of atomic vibrations, the analyses of energy transfer across interfaces can also be carried out in the frequency domain. From lattice dynamic point of view, the key factor that determines the phonon transport across two connecting materials is the overlapping of phonon PDOS between them.^{[62,63,73–75]} The overlapping/similarity of PDOS two connecting materials can be quantitatively analyzed using cosine similarity measure^[74]

where S is the value of similarity, and ) are the PDOSs of two connecting materials. For the MoS₂/Au interface, the PDOS of MoS₂ has notable phonon spectrum peaks in the range larger than 8 THz with a cut-off frequency of 15 THz, as shown in Fig. 8. However, for the Au atoms, the cut-off frequency is around 5 THz, and the peaks of its PDOS are in the range lower than 5 THz. Due to the different cut-off frequency between Au and MoS₂, there is an obvious mismatch in PDOS above 5 THz at the MoS₂/Au interface, which hinders the thermal transport across the interface and causes the interfacial thermal resistance.^[63]

Fig. 8.

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	Fig. 8. (color online) Phonon density of states of Au and MoS2 with different S vacancy concentration in the central region of MoS2. Reprinted with permission from Ref. [61], copyright (2016) by the Springer.

Besides MoS₂/metal edge contact, edge contact between TMDs and other 2D materials is also an interesting and important issue. For example, unlike low thermal conducting MoS₂, another 2D counterpart, graphene, has ultra-high thermal conductivity.^[76–79] Naturally, it is expected to construct edge-contacted MoS₂/graphene interface to speed up the heat transport. Figure 9 displays the schematic of MoS₂/graphene heterostructure with covalent-bonded interface. The ITC for this edge-contacted interface reaches , which is two orders higher than that of side-contacted MoS₂/graphene interface ( ).^[63,64] Thus this edge-contacting heterostructure will be one opportunity to overcome the bottleneck in thermal management of TMDs integrated devices.

Fig. 9.

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	Fig. 9. (color online) In-plane MoS2/graphene covalent-bonded heterostructure with hot and cold baths at the ends. Reprinted with permission from Ref. [63], copyright (2017) by the Springer.

According to the molecular dynamics simulation, it is found that each covalent bond at the interface serves as an independent channel for thermal conduction, which enables the ITC to be tuned linearly by adjusting the interfacial bond density.

The efficiency of different phonon modes transferring across the interface can be quantified by energy transmission coefficient α using the phonon wave packet method.^[81,82] The energy transmission coefficient α can be obtained by , where and A are the amplitudes of transmitted and initial phonon waves at the interface, respectively. Figure 10 shows the snapshots of a wave packet near the MoS₂/graphene interface. Figure 11 shows the energy transmission coefficient of phonon wave packets. It is found that the interface effectively blocks high frequency phonon modes due to the lack of LA and TA energy states in MoS₂ at the same frequencies as in graphene. For LA phonon mode, when the incident frequencies are lower than 7.3 THz, phonon transmission coefficient α keeps around 0.75, without remarkable change; however, it dramatically decreases, when the frequency reaches 8 THz. When the frequency is higher than 9 THz, the phonon wave packets are totally reflected.^[63] In contrast to LA phonon mode, the α of TA phonons is noticeably lower than that of LA phonons, even in the low frequency regime. With the same wave vector , for LA phonon wave packet, the α is about 0.75; while for TA phonon wave packet, α is only about 0.12.^[63] These results indicate that LA modes have the dominant contribution to the thermal transport across the interface.

Fig. 10.

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	Fig. 10. (color online) Snapshots of a phonon wave packet travelling across the MoS2/graphene interface. The travelling direction is indicated by blue arrows, and brown dashed lines denote the interface. (a) LA, , f = 1.90 THz. (b) LA, , f = 7.10 THz. (c) TA, , f = 1.18 THz. (d) LA, , f = 4.41 THz. Reprinted with permission from Ref. [63], copyright (2017) by the Springer.

Fig. 11.

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	Fig. 11. (color online) Energy transmission coefficient α of a phonon wave packet as a function of frequency. The inset displays the phonon dispersion of graphene and MoS2. The MoS2 phonon dispersion is adopted from Ref. [36] by first principle calculations. Reprinted with permission from Ref. [63], copyright (2017) by the Springer.

Meanwhile, the rapid development in materials synthesis techniques has made many novel edge-contacted in-plane 2D heterostructures possible. For example, recently, 2D graphene/h-BN in-plane heterostructures were successfully synthesized.^[82] In particular, it is now possible to grow atomically sharp interfaces between graphene and h-BN domains using combined atmospheric pressure chemical vapor deposition (CVD) and reactive ion etching.^[83] These fascinating advances in the synthesis of 2D heterostructures have laid a promising foundation for realizing such edge-contacted interfaces.

This article presents an overview of the thermal properties of TMDs in both experimental and theoretical perspectives. The lattice dynamics and thermodynamic properties of TMDs have been summarized. The experimental and theoretical studies reveal that the TMDs have relatively low thermal conductivities, which are 2–3 orders lower than that of graphene. The low thermal conductivities pose a significant challenge for efficient thermal management of TMDs-based devices. In the context of TMDs-based electro devices, heat dissipation from the devices and interconnects is primarily limited by their low thermal conductivities and the relatively weak van der Waals interfaces. The recent studies have suggested that this issue could be largely addressed by connecting TMDs and other materials with covalent bonds, and the interfacial thermal conductance of the covalent bonded interface could be highly tunable.