**2. Thermal conductivity**

surge of activity and demonstration of new physical phenomena. Despite its success, gra‐ phene still faces some severe problems in its nature of semi-metal or zero band-gap semicon‐ ductor and its incompatibility with the current Si-based semiconductor technology. Given that the honeycomb geometry is related to some of the exceptional properties of graphene, there is strong motivation to investigate whether changing carbon to other atom type might give rise to novel physical phenomena as well. An intuitive idea is to study similar 2D materials, such as siliceneandphosphorene.Actually, silicene,theSicounterpartofgraphene,cansolvetheabovementioned problems smoothly and thus has received intense interest lately. Given the fact that thermal transport plays a critical role in many applications such as heat dissipation in nanoe‐ lectronics and thermoelectric energy conversion, there has been an emerging demand in characterizing thermal (mainly phonons) transport property of silicene structures. Moreover, research results have shown that silicene exhibits a few novel thermal transport properties, which are fundamentally different from that of graphene, despite the similarity of their honeycomb lattice structure. Therefore, the abnormal physical property, primarily stemming from its unique low buckling structure, may enable silicene to open up entirely new possibili‐

With the state of the art, this book chapter aims to present theoretical investigations of thermal transport of broad 2D nanostructures in various forms, which have been carried out in our research group in the past few years. Heat transfer in such structures is not only directly relevant to optimizing the device performance such as improved thermal manage‐ ment for nanoelectronics and thermoelectric energy conversion efficiency, but also is a scientifically fundamental problem for many other similar two-dimensional systems (**Figure**

**Figure 1.** Comparison of the crystal structures (top view and side view) among (a) graphene, (b) silicene, and (c) phos‐ phorene. Graphene possesses perfect planar structure, silicene possesses low buckling structure, and phosphorene

ties for revolutionary electronic devices and energy conversion materials.

200 Two-dimensional Materials - Synthesis, Characterization and Potential Applications

**1**).

possesses pucker (hinge-like) structure.

Two-dimensional (2D) materials have been extensively studied in the past decade because of their novel physical and chemical properties [1, 2] and potential applications [3, 4]. For example, it has been found that graphene has extremely high thermal conductivity [5], which has great potential in the applications including electronic cooling and composite materials. In this section, we summarize the available experimental data and theoretical calculations on the phonon transport properties of several typical 2D materials consisting of only one type of element, including graphene, silicene, and phosphorene (**Figure 2**).

**Figure 2.** The phonon dispersion curves along the path passing through the main high-symmetry *k*-points in the irre‐ ducible Brillouin zone of graphene, silicene, and phosphorene. Three acoustic phonon modes (LA, TA, and FA) are labeled.

#### **2.1. Graphene**

Graphene, the first two-dimensional atomic crystal available to us, has attracted great attention due to its supreme mechanical, electronic, and optical properties. Besides these properties, thermal and thermoelectric properties of graphene are also very fascinating. Experiments have shown that the thermal conductivity of graphene reaches as high as about 3000 W/mK, which makes graphene very promising for thermal management applications such as heat dissipation in electronics [5]. In addition to the heat dissipation applications, due to its extremely high electrical conductivity, graphene has also been explored to be used as thermoelectric material by largely reducing its thermal conductivity by varieties of functionalization, such as con‐ structing graphene nanoribbons, hydrogenation, defects, and doping.

There have been a long time for an intriguing open question on the phonon transport in graphene: what is the relative contribution to heat conduction of LA, TA, and FA phonon polarization branches? The focused point is the importance of FA phonons from negligible to dominant. Klemens' theory states that thermal energy is mainly carried by longitudinal acoustic (LA) and transverse acoustic (TA) phonons, and contributions from out-of-plane flexural acoustic (FA) phonons are negligible due to their small group velocity in the Brillouin zone center and large Grüneisen parameter [6–8]. However, based on recent studies, the arguments for the dominant contributions of FA modes are made on the basis of the symmetrybased phonon-phonon scattering selection rule [9–11]. As for graphene, because of the reflectional symmetry of the structure, the phonon scattering processes involving odd number of FA modes are not allowed, which restricts the phase space for phonon-phonon scattering [9]. The suppression of scattering of FA leads to its extremely low scattering rate which is equivalent to a very large phonon lifetime, which results in the huge contribution to thermal conductivity from FA modes. The selection rule can be broken by placing graphene on any substrate or due to the presence of nanoscale corrugations. Therefore, it is intuitive to see that thermal conductivity of supported graphene is dramatically reduced.

#### **2.2. Silicene**

Silicene, the silicon counterpart of graphene, possesses a two-dimensional structure that leads to a host of interesting physical and chemical properties of significant utility. Compared to graphene, silicene is more compatible with silicon-based semiconductor devices and technol‐ ogies, and therefore has greater potential in nanoelectronic applications. In particular, in terms of thermoelectric application, silicene is even more exciting than graphene as the charge carrier of silicene is massless fermion, and the electrical conductivity of silicene is as high as that of graphene. At the same time, the thermal conductivity of silicene is expected to be much lower than that of graphene due to its buckled atomic structure. In addition to thermoelectric applications, silicene, with supreme electronic properties similar to those of graphene, has also shown great potential for other applications, such as nanoelectronics. For example, in contrast to graphene, the buckled atomic structure breaks the symmetry of the silicene, making it possible to open a band gap by applying electric field [12–14]. Monolayer silicene has been successfully fabricated on substrates such as Ag(1 1 0) [15], Ir(1 1 1) [16], and Ag(1 1 1) [17] surfaces. Recently, Tao et al. have demonstrated silicene-based transistors can operate at room temperature [4]. Although the performance is still moderate and the lifetime of this transistor is only a few minutes, it has attracted significant research interest in silicene-based devices [18– 20]. Therefore, there is an urgent demand to quantify the thermal transport property of silicene, and it is of great interest to explore the role of buckled lattice on phonon transport mechanisms compared with the planar graphene.

**2.1. Graphene**

**2.2. Silicene**

Graphene, the first two-dimensional atomic crystal available to us, has attracted great attention due to its supreme mechanical, electronic, and optical properties. Besides these properties, thermal and thermoelectric properties of graphene are also very fascinating. Experiments have shown that the thermal conductivity of graphene reaches as high as about 3000 W/mK, which makes graphene very promising for thermal management applications such as heat dissipation in electronics [5]. In addition to the heat dissipation applications, due to its extremely high electrical conductivity, graphene has also been explored to be used as thermoelectric material by largely reducing its thermal conductivity by varieties of functionalization, such as con‐

There have been a long time for an intriguing open question on the phonon transport in graphene: what is the relative contribution to heat conduction of LA, TA, and FA phonon polarization branches? The focused point is the importance of FA phonons from negligible to dominant. Klemens' theory states that thermal energy is mainly carried by longitudinal acoustic (LA) and transverse acoustic (TA) phonons, and contributions from out-of-plane flexural acoustic (FA) phonons are negligible due to their small group velocity in the Brillouin zone center and large Grüneisen parameter [6–8]. However, based on recent studies, the arguments for the dominant contributions of FA modes are made on the basis of the symmetrybased phonon-phonon scattering selection rule [9–11]. As for graphene, because of the reflectional symmetry of the structure, the phonon scattering processes involving odd number of FA modes are not allowed, which restricts the phase space for phonon-phonon scattering [9]. The suppression of scattering of FA leads to its extremely low scattering rate which is equivalent to a very large phonon lifetime, which results in the huge contribution to thermal conductivity from FA modes. The selection rule can be broken by placing graphene on any substrate or due to the presence of nanoscale corrugations. Therefore, it is intuitive to see that

Silicene, the silicon counterpart of graphene, possesses a two-dimensional structure that leads to a host of interesting physical and chemical properties of significant utility. Compared to graphene, silicene is more compatible with silicon-based semiconductor devices and technol‐ ogies, and therefore has greater potential in nanoelectronic applications. In particular, in terms of thermoelectric application, silicene is even more exciting than graphene as the charge carrier of silicene is massless fermion, and the electrical conductivity of silicene is as high as that of graphene. At the same time, the thermal conductivity of silicene is expected to be much lower than that of graphene due to its buckled atomic structure. In addition to thermoelectric applications, silicene, with supreme electronic properties similar to those of graphene, has also shown great potential for other applications, such as nanoelectronics. For example, in contrast to graphene, the buckled atomic structure breaks the symmetry of the silicene, making it possible to open a band gap by applying electric field [12–14]. Monolayer silicene has been successfully fabricated on substrates such as Ag(1 1 0) [15], Ir(1 1 1) [16], and Ag(1 1 1) [17] surfaces. Recently, Tao et al. have demonstrated silicene-based transistors can operate at room

structing graphene nanoribbons, hydrogenation, defects, and doping.

202 Two-dimensional Materials - Synthesis, Characterization and Potential Applications

thermal conductivity of supported graphene is dramatically reduced.

The intrinsic physical properties of silicene, such as lattice thermal conductivity, have been an active area of research. Although the thermal conductivity of silicene has not been measured in experiments due to the difficulty of synthesizing freestanding silicene, several numerical simulations have predicted the thermal conductivity of silicene and the results at 300 K range from 5 to 69 W/mK [21–23]. Most of the numerical simulations are based on classical molecular dynamics and the discrepancy of results mainly arises from the different interatomic interac‐ tion potentials used. Notably, first-principles-based lattice dynamics predicted that the thermal conductivity of silicene is in the range of 20–30 W/mK [23, 24], which should be more reliable.

From the aspect of theoretical study, the widely used classical molecular dynamics simulation is an appropriate way to investigate the transport phenomena and mechanisms in silicene. Based on the optimized SW potential of SW1 and SW2 obtained by Zhang et al. [21], the thermal conductivity of silicene calculated from non-equilibrium molecular dynamics (NEMD) is 8.64 W/mK (SW1) and 11.77 W/mK (SW2) at 230 K. One would notice that the results for the equilibrium molecular dynamics (EMD) and NEMD methods are not the same, even after the thermal conductivity was extrapolated to infinitely long in the case of NEMD simulation. Nevertheless, the results confirm that the thermal conductivity of monolayer silicene is ultralow, which is one order of magnitude lower than the value of bulk silicon (about 150 W/mK from experiments).

The thermal conductivity was also calculated by Xie et al. using the single-mode relaxation time (SMRT) model derived from Boltzmann transport equation (BTE), where the scattering of certain phonon mode is irrelevant to the condition of other phonon modes, but other phonon modes are assumed to stay in their equilibrium condition. The interactions for silicon atoms are described by the SW1/SW2 potentials, and the single-point energies are calculated using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package. The thermal conductivity values at 300 K are 3.33 W/mK for the optimized SW1 parameters and 5.43 W/mK for the optimized SW2 parameters. The thermal conductivity calculated by anharmonic lattice dynamics (ALD) is smaller than that calculated by molecular dynamics (MD) simulations. The discrepancy was argued to be mainly attributed to the strong normal scattering near the zone center in two-dimensional materials, as also seen in graphene. After that, first-principles-based ALD method is combined with BTE to accurately calculate the lattice thermal conductivity of silicene by Xie et al. [24]. The main difference from the previous study is that the interactions among atoms are described based on first-principles calculations, and the accuracy should be much higher. The temperature dependence of thermal conductivity was explicitly considered and mode-specific contribution to overall thermal conductivity was analyzed and discussed. Due to the buckling structure of silicene, the flexural mode is not purely out-of-plane vibration, but also contains small components in the in-plane directions. The intrinsic lattice thermal conductivity of silicene is 9.4 W/mK at 300 K. Later on, due to the development of the calculation method, the BTE can be iteratively solved, where the thermal conductivity becomes unbounded and the iterative procedure reflects the process of the redistribution of phonon modes driven by the heat flux [11, 25–27]. However, the underlying mechanisms are still not well understood. Based on the iterative solution of BTE, the thermal conductivity of silicene is in the range of 15–30 W/mK for three finite sizes of 0.3, 3, and 30 μm [28]. Gu and Yang and Kuang et al. also calculated the thermal conductivity of silicene using the first-principles-based BTE approach, and obtained similar magnitude of thermal conduc‐ tivity [23, 29].

## **2.3. Phosphorene**

Black phosphorus (BP) is another emerging material which has a puckered layered structure with intra-layer interaction dominated by van der Waals forces [30–32]. Few-layer BP has recently been successfully fabricated by mechanical exfoliation [33, 34] and has generated tremendous interest among scientists [33–44]. Phosphorene, the single layer counterpart of BP, is another interesting 2D structure with high carrier mobility proved by experiments [33, 34, 36] and a large fundamental direct band gap (~1.5 eV) [39], which makes BP promising for lots of nanoelectronic applications [32]. For example, some previous theoretical and experimental works have illustrated that phosphorene can be used as nanoelectronic devices, such as fieldeffect transistors and photo-transistors [33, 34, 36–38]. Besides the extensive studies on its electrical properties, there are also a lot of explorations on its potential applications in ther‐ moelectrics [35, 42, 43]. All these electrical and thermoelectrical applications are closely related to thermal properties. Considering the potential valuable applications of phosphorene as nano-/opto-electronic and thermoelectric devices, it is necessary to fundamentally study the phonon transport properties of this new 2D material.

There have been some experiment works on the measurement of the thermal conductivity of bulk BP [45] and phosphorene films with different thicknesses [46–48]. The thermal conduc‐ tivity of monolayer phosphorene was also reported theoretically by several independent groups using various methods, such as analytical estimations [35, 49], MD simulation with optimized Stillinger-Weber potential [50–52], relaxation time approximation (RTA) [45, 53– 55], and iterative method for solving BTE [45, 54]. However, the exact value of the lattice thermal conductivity of monolayer phosphorene is still unclear, since these results are even one order of magnitude different from each other. For example, the thermal conductivity of monolayer phosphorene along zigzag direction ranges from 30 W m−1 K−1 to 152.7 W m−1 K−1, while that along armchair direction ranges from 9.9 W m−1 K−1 to 63.9 W m−1 K−1 [35, 50–52]. The huge deviation might be due to the different calculation methods or parameters. For example, Jain et al. calculated the thermal conductivity using similar iterative method based on ALD/BTE as employed in this work, and the test for the convergence of thermal conductivity was performed with respect to the interactions cutoff [54]. However, the thermal conductivity does not show a distinct convergence trend versus the interactions cutoff while jumps in a quite wide range, which might lie in that: (1) the lacking of long-range electrostatic interactions in their work, which are taken into account by adding to the dynamical matrix a correction from dielectric tensor and Born effective charges; (2) the scalar relativistic pseudopotential used in their work does not involve the van der Waals interactions, which are considered having significant effect on the properties of bulk BP and phosphorene [31, 56]. In another work by Zhu et al. [53] based on RTA, the interactions range of third-order Interatomic force constants (IFCs) is truncated up to 4.4 Å, with which the interaction is only taken into account up to 7th nearest neighbors. Therefore, to end the confusing situation, it is necessary to perform systematic study to precisely quantify the phonon transport properties of phosphorene, which would be of significance to its further applications in nano-/opto-electronics and thermoelec‐ trics (**Figure 3**).

The intrinsic lattice thermal conductivity of silicene is 9.4 W/mK at 300 K. Later on, due to the development of the calculation method, the BTE can be iteratively solved, where the thermal conductivity becomes unbounded and the iterative procedure reflects the process of the redistribution of phonon modes driven by the heat flux [11, 25–27]. However, the underlying mechanisms are still not well understood. Based on the iterative solution of BTE, the thermal conductivity of silicene is in the range of 15–30 W/mK for three finite sizes of 0.3, 3, and 30 μm [28]. Gu and Yang and Kuang et al. also calculated the thermal conductivity of silicene using the first-principles-based BTE approach, and obtained similar magnitude of thermal conduc‐

204 Two-dimensional Materials - Synthesis, Characterization and Potential Applications

Black phosphorus (BP) is another emerging material which has a puckered layered structure with intra-layer interaction dominated by van der Waals forces [30–32]. Few-layer BP has recently been successfully fabricated by mechanical exfoliation [33, 34] and has generated tremendous interest among scientists [33–44]. Phosphorene, the single layer counterpart of BP, is another interesting 2D structure with high carrier mobility proved by experiments [33, 34, 36] and a large fundamental direct band gap (~1.5 eV) [39], which makes BP promising for lots of nanoelectronic applications [32]. For example, some previous theoretical and experimental works have illustrated that phosphorene can be used as nanoelectronic devices, such as fieldeffect transistors and photo-transistors [33, 34, 36–38]. Besides the extensive studies on its electrical properties, there are also a lot of explorations on its potential applications in ther‐ moelectrics [35, 42, 43]. All these electrical and thermoelectrical applications are closely related to thermal properties. Considering the potential valuable applications of phosphorene as nano-/opto-electronic and thermoelectric devices, it is necessary to fundamentally study the

There have been some experiment works on the measurement of the thermal conductivity of bulk BP [45] and phosphorene films with different thicknesses [46–48]. The thermal conduc‐ tivity of monolayer phosphorene was also reported theoretically by several independent groups using various methods, such as analytical estimations [35, 49], MD simulation with optimized Stillinger-Weber potential [50–52], relaxation time approximation (RTA) [45, 53– 55], and iterative method for solving BTE [45, 54]. However, the exact value of the lattice thermal conductivity of monolayer phosphorene is still unclear, since these results are even one order of magnitude different from each other. For example, the thermal conductivity of monolayer phosphorene along zigzag direction ranges from 30 W m−1 K−1 to 152.7 W m−1 K−1, while that along armchair direction ranges from 9.9 W m−1 K−1 to 63.9 W m−1 K−1 [35, 50–52]. The huge deviation might be due to the different calculation methods or parameters. For example, Jain et al. calculated the thermal conductivity using similar iterative method based on ALD/BTE as employed in this work, and the test for the convergence of thermal conductivity was performed with respect to the interactions cutoff [54]. However, the thermal conductivity does not show a distinct convergence trend versus the interactions cutoff while jumps in a quite wide range, which might lie in that: (1) the lacking of long-range electrostatic interactions in their work, which are taken into account by adding to the dynamical matrix a correction

tivity [23, 29].

**2.3. Phosphorene**

phonon transport properties of this new 2D material.

**Figure 3.** The comparison of the thermal conductivity obtained from experiments and theoretically reported results. The left box shows the thermal conductivity of monolayer phosphorene from different methods [35, 45, 49–55]. The middle box shows the thermal conductivity of phosphorene films with different thicknesses from experiments [46–48]. The right box shows the thermal conductivity of bulk BP from both experiment and theoretical calculation [45].

The thermal conductivity of monolayer phosphorene is in the same order of magnitude as that of silicene (~20 W m−1 K−1) [21–24, 28], while two orders of magnitude lower than that of graphene (3000–5000 W m−1 K−1) [9, 11]. The thermal conductivity is mainly contributed by phonon modes below the gap which separates the phonon branches into two regions with each region containing six branches. The phonon modes in the region below the gap contribute more than 85% to the thermal conductivity, and dominate the anisotropy. The thermal conductivity along armchair direction is mainly contributed by LA, while along zigzag direction, TA/FA also contribute a lot to the thermal conductivity besides LA. The contribu‐ tions to thermal conductivity along zigzag and armchair directions from FA are 16% and 8% at room temperature, respectively, which is close to that of silicene (7.5%) but much smaller than that of graphene (75%) [11, 23]. As for graphene, because of the reflectional symmetry of the structure, the phonon scattering processes involving odd number of FA modes are not allowed, which is the so-called symmetry-based phonon-phonon scattering selection rule [9]. Compared with graphene, the symmetry-based phonon-phonon scattering selection rule is broken by the hinge-like structure of phosphorene, resulting in a large scattering rate of FA, which thus leads to its small contribution to the thermal conductivity. Note that FA contributes to the thermal conductivity along zigzag direction more than that along armchair direction. The reason might lie in the feature of the hinge-like structure that it is more "uneven" along armchair direction because of the up and down of the sublayers chains.
