*6.1. Graphene*

As the first 2D material successfully prepared, graphene became the favorite of scientific research once discovered. It has a series of excellent physical properties including ultra-high conductivity, ultra-high carrier mobility, etc. [16,130,131]. In terms of thermal conductivity, graphene also performs well, and its intrinsic thermal conductivity at room temperature can reach 2000–3000 W/m·K, which is the highest thermal conductivity material found so far. In 2008, for the first time, Balandin et al. [40] measured the thermal conductivity of suspended single-layer graphene at room temperature using the Raman method and graphite bulk materials reaching 4840–5300 W/m·K; however, upon further study by scientists, it was discovered that the experiment may have had an excessive estimate of the Raman laser absorption power of graphene, resulting in a result 4–6 times larger [30,132]. In 2010, Wei et al. [133] used the same method to measure the thermal conductivity of suspended single-layer graphene and, by employing a laser power meter, measured the Raman laser absorption rate of graphene simultaneously.

The results illustrate that the thermal conductivity of single-layer graphene grown by chemical vapor deposition (CVD) is around 2500–3100 W/m·K (*T* = 350 K) and 1200–1400 W/m·K (*T* = 500 K). It has also been found that the shape, size, and measurement environment of the suspended part of the graphene will affect the final result. In addition to the dispute about the laser absorption rate, it caused the difference between different experimental results. Another major reason for the large difference is that the preparation methods of graphene are different (mechanical exfoliation or chemical vapor deposition), resulting in certain differences in its quality (impurities, grain boundaries, organic residues, etc.). These factors will affect the phonon and generate additional scattering. Table 1 lists the thermal conductivity of suspended single-layer/multi-layer graphene measured by different experimental methods, including the Raman method.


**Table 1.** Experimental detail of thermal conductivity in suspended single-/few-layer graphene from different studies.


**Table 1.** *Cont.*

In order to understand graphene's ultra-high thermal conductivity, it is important to know how different phonon modes contribute to it. In graphene, heat is mainly transmitted through acoustic phonons, and its acoustic phonon modes include in-plane acoustic longitudinal wave (LA), in-plane acoustic transverse wave (TA), and out-of-plane acoustic shear wave (ZA). Nika et al. [140] believe that the heat transport process in single-layer graphene is almost entirely carried by the LA/TA phonon mode, while the contribution of the ZA phonon mode to the thermal conductivity is negligible. However, according to Lindsay et al. [52,53], the ZA phonon mode has a relatively high density of low-frequency phonons, so its contribution to the thermal conductivity will be relatively large, and it is predicted that the contribution of the ZA phonon mode to the thermal conductivity of single-layer graphene can reach about 70% at room temperature. In addition to ultra-high thermal conductivity, graphene is also an excellent platform for studying phonon ballistic transport. Because the mean free path of phonons in graphene is very long, it can reach the order of micrometers at room temperature, so the (quasi) ballistic transport properties of phonons can be studied by adjusting the length and width of the graphene band [141]. In 2010, Xu Xiangfan et al. used the suspended thermal bridge technique to measure the monolayer for the first time. The thermal conductivity of CVD graphene nanoribbons (width of about 3 μm, length of about 500 nm) is found to have a certain exponential relationship between thermal conductivity and temperature in the low-temperature region (T < 140 K), which can be explained by the thermal transport in single-layer suspended graphene, mainly depends on the ZA phonon mode, and experimental evidence of phonon quasi-ballistic transport has been found [97,138]. Although Petters et al. [98] object to the low thermal conductivity (about 225 W/m·K) measured in the above experiment, it is believed that the observed *k*~1.53 ± 0.18 is not from ZA phonons but due to impurities on the surface of the sample.

The thermal conductivity theory discussed above is calculated based on the threephonon scattering model. Four-phonon scattering is often directly ignored because it is a type of phonon scattering behavior that only appears in high-temperature regions [142]. However, recent studies found that even at room temperature, the four-phonon scattering process caused by a large number of low-energy ZA mode phonons in single-layer graphene cannot be ignored. Therefore, the thermal conductivity of single-layer graphene, calculated using only the three-phonon scattering model, may be relatively high [142,143]. By solving the Boltzmann equation and introducing four-phonon scattering, Feng et al. [142] found that the thermal conductivity of single-layer graphene at room temperature is only 810 W/m·K; this value is much lower than the calculation result including only three-phonon scattering (about 3383 W/m·K), and the result shows that under the influence of four-phonon scattering, the contribution of the ZA phonon mode to the thermal conductivity is only 31%. The atomic force constant used in this work to calculate the thermal conductivity of single-layer graphene at room temperature is 0 K. Subsequently, Gu Xiaokun et al. [143]

corrected the above results, and the result value obtained was slightly larger than that in the above literature, further confirming the severe impact of four-phonon scattering on the thermal conductivity of single-layer graphene at room temperature. It can be seen that the thermal conductivity of single-layer-suspended graphene is still inconclusive. There is a certain controversy about this problem, whether in theory or in experiments. In practical applications, graphene is more likely to be attached to a certain substrate.

Therefore, in addition to floating graphene, the in-plane heat conduction of graphene on substrate performance is also necessary to study. Seol et al. [52] measured the in-plane thermal conductivity of a single layer of graphene on a silicon oxide substrate (about 600 W/m·K, 300 K). When the graphene is attached to the substrate, the ZA phonon mode will be suppressed, so its in-plane thermal conductivity is lower than that of suspended graphene. As shown in the above theoretical calculation results, the contribution of the ZA phonon to the thermal conductivity can reach about 70% at room temperature, but when there is a lining at the end, the contribution of the ZA phonon mode will be severely suppressed or even disappear, and the thermal conductivity value will be reduced from ~3000 W/m·K to ~600 W/m·K. This experiment effectively verifies that the graphene ZA phonons make a significant contribution to its thermal conductivity. In addition to silicon oxide, silicon nitride is also a common substrate material. Thong et al. [103] measured the in-plane thermal conductivity of multi-layer graphene on a silicon nitride substrate. The value is ~150–1250 W/m·K (room temperature). In order to further verify the contribution of ZA phonons to thermal conductivity, Wang et al. [99] deposited gold atoms on the surface of three-layer suspended graphene. It was found that its thermal conductivity decreased from about 1500 W/m·K to about 270 W/m·K (a decrease of 82%) and Seol et al. came to the same conclusion. Jang et al. [144] studied the heat transport properties of the SiO2−graphene −SiO2 sandwich structure and found that the thermal conductivity will be further reduced, especially the single-layer sandwich graphene structure. Iits room temperature thermal conductivity is far below 160 W/m·K (the specific value in this article is too low to be measured and only an upper limit is given), indicating that the substrate has a very significant inhibitory effect on the thermal conductivity of graphene.

#### *6.2. Boron Nitride*

Due to the large bandgap and the very smooth surface, boron nitride (h-BN) is an ideal type of dielectric material. At the same time, the thermal conductivity of boron nitride bulk materials (about 400 W/m·K, room temperature) is very close to copper, and its mass is much lower than copper under the same volume, so it has broad application prospects in terms of the heat dissipation of electronic devices [145,146]. Boron nitride is called white graphene. The crystal structure is similar to graphene. Nitrogen atoms and boron atoms in the plane are interlaced to form a honeycomb structure, and the layers are combined with each other by van der Waals forces. It is one of the two-dimensional materials discovered earlier [147]. The physical properties of boron nitride and graphene also have certain similarities. Lindsay et al. [148] predicted the room temperature inplane thermal conductivity of single-layer boron nitride through theoretical research by solving the Boltzmann equation. The rate is 600 W/m·K, which is higher than that of the boron nitride bulk material. At the same time, it is also found that the contribution of the out-of-plane ZA phonon mode to the thermal conductivity can reach ~60%. In 2013, Jo et al. [32] adopted a microbridge resistance thermometer method to measure the in-plane thermal conductivity of multi-layer boron nitride (250 W/m·K, five layers; 360 W/m·K, 11 layers; *T* = 300 K). They believe that the reason why the measured data is lower than the theoretical prediction and even lower than the thermal conductivity of the boron nitride bulk material is mainly due to a large amount of organic residue on the surface of the boron nitride during the experiment, which causes serious phonon scattering. In the subsequent experiments of the thermal conductivity of the multi-layer boron nitride floating plane, it was not observed that the thermal conductivity exceeds the bulk material, and the quality of the sample is the key factor [108,113,149]. Wang et al. [100] improved the

sample transfer method (PDMS-assisted dry transfer), which greatly reduced the organic residues on the surface of the boron nitride film. At the same time, they used high-quality bulk materials to mechanically peel off the multi-layer boron nitride and for the first time, measured the thermal conductivity of the suspended double-layer boron nitride as being greater than that of the bulk material, reaching 460–625 W/m·K at room temperature. Subsequently, Cai et al. [109] used the Raman method to measure the thermal conductivity of single-layer/double-layer and three-layer floating boron nitride near room temperature. The thermal conductivity value decreases as the thickness increases, but it is still higher than bulk boron nitride. The thermal conductivity of single-layer boron nitride reaches ~751 W/m·K. The highest value of the thermal conductivity of single-layer/multi-layer boron nitride was obtained in previous experiments. Table 2 lists the in-plane thermal conductivity of floating boron nitride measured in different documents. According to the data in the table, the thermal conductivity of multi-layer boron nitride prepared by the mechanical exfoliation method is generally higher than that prepared by the CVD method. The reason is that the chemical vapor deposition method often introduces more in the sample. However, the mechanical exfoliation method performs better at controlling the sample quality.

**Table 2.** Experimental results of thermal conductivity of suspended single-/few-layer h-BN in different kinds of studies.


#### *6.3. Molybdenum Sulfide and Other Transition Metal Sulfides*

Transition metal sulfides (MX2, where M is transition metal elements such as Mo, W, Ti, and X represents chalcogen elements, including S, Se, and Te) are a very important group of two-dimensional materials and their crystal structure is a "sandwich"-like layered structure [150]. Unlike single-layer graphene, single-layer boron nitride, and other twodimensional materials that only contain one atomic layer, a single-layer transition metal sulfide contains three atomic layers (the transition metal atomic layer is sulfurized). The atomic layer of group elements is "sandwiched" in the middle, as shown in Figure 9.

Molybdenum sulfide is the most widely studied transition metal sulfide. Because of its controllable bandgap and excellent electrical properties, it can also exist stably in the air. It is considered a material for the next generation of microelectronic devices with grea<sup>t</sup> potential. Optics, thermoelectrics, and other fields also have certain application prospects [151–153]. Sahoo et al. [112] used the Raman method to measure the in-plane thermal conductivity of 11 layers of molybdenum sulfide (about 52 W/m·K, room temperature). Later, Yan et al. [56] and Jo et al. [129] measured the in-plane thermal conductivity of single-layer and multi-layer molybdenum sulfide at room temperature and the values were 35–52 W/m·K. However, Zhang et al. [154] also used the Raman method to measure the

in-plane thermal conductivity of single-layer/double-layer molybdenum sulfide, and the result (77–84 W/m·K) was much larger than the previous experimental data. It is because the critical data, such as the relationship between the Raman peak frequency change and temperature of molybdenum sulfide, the absorption power of the Raman laser, and the contact thermal resistance obtained in the experiment are quite different from the previous literature. Aiyiti et al. [104] used the electron beam self-heating method to measure the in-plane thermal conductivity of the multi-layer molybdenum sulfide. This is the first time this method has been applied to the experimental measurement of the thermal conductivity of two-dimensional materials. The experimental results also confirm the feasibility of this method. The results of the experimental measurements of the in-plane thermal conductivity of molybdenum sulfide are summarized in Table 3.

**Figure 9.** Molybdenum sulfide [151]. Reprinted with permission from ref. [10]. AIP Publishing and Copyright Clearance Center.

**Table 3.** Experimental results of thermal conductivity of single-/few-layer MoS2 in different kinds of studies.


Compared with graphene and boron nitride, the crystal structure of molybdenum sulfide has certain differences, so its thermal conductivity properties will also be different, mainly as follows: First, the thermal conductivity of single-layer molybdenum sulfide is higher than that of single-layer graphene and single-layer nitrogen. The boron sulfide is 1-2 orders of magnitude lower. Wei et al. [35] found through theoretical calculations that the low thermal conductivity of single-layer molybdenum sulfide is due to the lower phonon group velocity and the larger Grüneisen constant. As a result, the mean free path of phonons is only 14.6 nm. Secondly, in single-layer graphene and single-layer boron nitride, the contribution of the out-of-plane ZA phonon mode to the thermal conductivity is more than 50%, but in a single layer of molybdenum sulfide, the contribution of the in-plane phonon mode to the thermal conductivity exceeds the out-of-plane phonon mode. Finally, unlike graphene and boron nitride, single-/multi-layer molybdenum sulfide has been experimentally measured. The in-plane thermal conductivity is lower than the in-plane thermal conductivity of molybdenum sulfide block (85–110 W/m·K, room temperature). This phenomenon is inconsistent with the theoretical prediction. Gu et al. [57] predicted that the room temperature in-plane thermal conductivity of a single layer of molybdenum sulfide could reach 138 W/m·K. The reason may be that the quality of molybdenum sulfide in these experiments has not reached a good condition, or the deeper reasons need to be further studied. In addition to the in-plane thermal conductivity, the inter-plane thermal conductivity of molybdenum sulfide is also one of the issues worthy of study, but there are relatively few studies in this direction. Initially, Muratore et al. [159] and Cahill et al. [160] measured the room temperature interfacial thermal conductivity of bulk molybdenum sulfide, which was only 2–3 W/m·K. However, Jiang et al. [93] showed a higher numerical result (about 4.75 W/m·K) experimentally, and the result is closer to the theoretical prediction. The difference between the above experimental results is in the follow-up Sood et al. [92] explained in the experiment. They measured the room-temperature inter-plane thermal conductivity of different thicknesses of molybdenum sulfide by the TDTR method, and the results showed that the room-temperature inter-plane thermal conductivity of samples with a thickness of 240 nm and 20 nm were 2.0 ± 0.3 W/m·K, 0.9 ± 0.2 W/m·K. Through comparison with the above experimental data, it is found that as the thickness increases, the inter-face thermal conductivity of molybdenum sulfide increases, and when the thickness reaches about 1 μm, the inter-face thermal conductivity value gradually approaches the saturation threshold (about 5 W/m·K). Theoretical calculation results show that the mean free path of phonons between the molybdenum sulfide surfaces far exceeds the previously estimated value (1.5–4 nm), and more than 80% of the heat transport is contributed by phonons with a mean free path of 10–500 nm.

With the continuous development of the preparation technology of two-dimensional materials, more and more multi-layer transition metal sulfides have been discovered, so their thermal conductivity is gradually being studied. Table 4 lists other transitions in different literatures except for molybdenum sulfide. Experimental measurement of the thermal conductivity of metal sulfides in the suspended plane. From the data in the table, it can be seen that although the crystal structures of these materials are very similar, their thermal conductivity properties are significantly different.


**Table 4.** Experimental detail of thermal conductivity of single-/few-layer transition metal dichalcogenides in different literatures.

#### *6.4. Black Phosphorus, Black Arsenic*

Due to the advantages of the controllable bandgap and relatively high switching, black phosphorous (BP) is one of the first materials for the next generation of microelectronic devices to be studied [168–170]. However, initially, researchers were interested in the thermal conductivity of black phosphorous. This is mainly because of its in-plane anisotropic "Great Wall"-like structure [128], which may lead to the anisotropy of thermal conductivity [171]. It is worth noting that the black phosphorous pole is easy to oxidize, so in experiments with black phosphorus, the exposure time of the sample in the air needs to be strictly controlled. Qin et al. [172] predicted the room temperature surface of the single-layer black phosphorus along the Zigzag (ZZ) direction and the (Armchair) AC direction through theoretical research. The internal thermal conductivity ratio can reach 30.15 W/m·K in the ZZ direction and 13.65 W/m·K in the AC direction, and due to its "Great Wall"-like structure, the out-of-plane phonon mode has a positive effect on the thermal conductivity. The contribution of efficiency is very low (about 5%). Lou et al. [110] experimentally measured the in-plane thermal conductivity of the multi-layer black phosphorus with different thicknesses at room temperature and the smallest thickness was ~10 nm. The thermal conductivity in the ZZ direction is 20 W/m·K, while the thermal conductivity in the AC direction is only ~10 W/m·K, which confirms the above theoretical prediction. As the same main group element of phosphorus, arsenic, i.e., black arsenic (Bas), has a crystal struc-

ture similar to black phosphorus and also has a significant in-plane thermal conductivity anisotropy effect. Chen et al. [173] first experimentally measured the in-plane thermal conductivity of black arsenic with a thickness of 124 nm along with the ZZ and AC directions (5 W/m·K, ZZ direction; 3 W/m·K, AC direction, 350 K). In subsequent experiments, researchers measured the in-plane thermal conductivity of multi-layer black phosphorus with different thicknesses using the Raman method, thermal bridge method, etc. They found the anisotropy of in-plane thermal conductivity. In these experiments, the thickness of the multi-layer black phosphorus was above 10 nm. This is because the chemical properties of black phosphorus are too active and it is challenging to prepare single-layer black phosphorus in heat conduction experiments. Therefore, the heat conduction properties of single-layer or few-layer black phosphorus need to be further studied.
