Graphene versus MoS{}_{\text{2}}: A short review

Abstract

Graphene and MoS₂ are two well-known quasi two-dimensional materials. This review presents a comparative survey of the complementary lattice dynamical and mechanical properties of graphene and MoS₂, which facilitates the study of graphene/MoS₂ heterostructures. These hybrid heterostructures are expected to mitigate the negative properties of each individual constituent and have attracted intense academic and industrial research interest.

Graphical abstract

Keywords

graphene / molybdenum disulphide / lattice dynamics / mechanical properties

1 Introduction

Quasi-two-dimensional (Q2D) materials have many novel properties and have attracted intense research interest over the past decades. The Q2D family of materials keeps expanding. The Q2D family currently contains the following materials: graphene, hexagonal boron nitride, two-dimensional (2D) honeycomb silicon, layered transition metal dichalcogenides (including MoS₂ and WS₂), black phosphorus, and 2D ZnO. Among the Q2D family of materials, graphene is the most well known. Novoselov and Geim were awarded the Nobel Prize in physics for graphene in 2010 [1].

The investigations on graphene are extensive but not exhaustive [2]. Nevertheless, these studies provide helpful guidance for understanding the whole Q2D family because many of the experimental set ups (initially for graphene) can be used to perform measurements on other materials in this family. For example, the mechanical properties of single-layer MoS₂ (SLMoS₂) were successfully measured using the same nanoindentation platform as graphene [3, 4]. In the theoretical community, many theorems or approaches, which were initially developed to study graphene, are applicable to other Q2D materials. Some of these extensions may be trivial because of the common two-dimensional nature of these materials. However, the extensions may enable new findings as the Q2D materials have different microscopic structures. For example, the bending modulus of SLMoS₂ can be derived using a similar analytic approach as that used for graphene even though the bending modulus of SLMoS₂ is about seven times larger than that of graphene, owing to its trilayer structure (one Mo layer sandwiched between two S layers) [5–9]. Another example is the puckered micro structure of black phosphorus, which leads to a negative Poisson’s ratio in the out-of-plane direction [10].

As a result, graphene has attracted ongoing research interest from both the academic and applied communities. Many review articles have been devoted to graphene [1, 11–19]. In addition, researchers have begun examining possible applications of other Q2D materials, using the knowledge gained from graphene. In particular, MoS₂ has attracted considerable research interest, and many review articles have been published on MoS₂ [20–24].

The present review provides a detailed comparison of the mechanical properties of graphene and SLMoS₂ to clarify the positive and negative properties of the individual materials and to highlight the possible advanced features and drawbacks of graphene/MoS₂ heterostructures [25]. These hybrid heterostructures are expected to mitigate the negative properties of each individual constituent material. For example, graphene/ MoS₂/graphene heterostructures have efficient photon absorption and electron-hole creation properties because of the enhanced light-matter interactions in the SLMoS₂ layer [25]. Another experiment recently showed that MoS₂ can be protected from radiation damage with graphene layer coatings [26], exploiting the outstanding mechanical properties of graphene.

In this review, we introduce and compare the following properties for graphene and SLMoS₂: the structure, interatomic potential, phonon dispersion, Young’s modulus, yield stress, bending modulus, buckling phenomenon, nanomechanical resonator, thermal conductivity, electronic band structure, and optical absorption. We further discuss the properties of the graphene/MoS₂ heterostructure. This article concludes with a table listing the major results for all properties compared in the article.

2 Structure and interatomic potential

Structure. Figure 1(a) shows that graphene has a honeycomb lattice structure with a D_{6h} point group. There are two nonequivalent carbon atoms in the unit cell. These two carbon atoms are reflected onto each other by the inverse symmetry operation from the D_{6h} point group. The lattice constant is a=2.46 Å, and the C-C bond length is b=a/\sqrt{3}=1.42 Å [27].

Figure 1(b) shows the top view of the SLMoS₂ structure, which is a trilayer structure with one Mo atomic layer sandwiched between two S atomic layers. The small yellow spheres represent the projection of the two outer S atomic layers onto the Mo atomic layer. The point group for SLMoS₂ is D_{3h}. The R_{\pi } rotation symmetry is broken in SLMoS₂. There are two S atoms and one Mo atom in the unit cell. It should be noted that one of the S atoms is not visible due to the trilayer structure. The lattice constant for the in-plane unit cell is a=3.09 Å, and the Mo-S bond length is b=2.39 Å. These values were computed using the Stillinger-Weber (SW) potential [28]; they agree with both the first-principles calculations [29] and the experiments [30].

Interatomic potential. The interactions between the carbon atoms in graphene can be calculated using four different computation cost levels. The first-principles calculation is the most expensive approach for computing the interatomic energy of graphene. Many existing simulation packages can be used for such calculations, including the commercially available Vienna Ab-initio Simulation Package (VASP) [31] and the freely available Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA) package [32]. To reduce the computational cost, Brenner et al. developed an empirical potential for carbon-based materials, including graphene [33]. The Brenner potential takes the form of the bond-order Tersoff potential [34] and is able to capture most of the linear properties and many of the nonlinear properties of graphene. For instance, the Brenner potential can describe the formation and breakage of bonds in graphene, providing a good description of its structural, mechanical, and thermal properties. The Tersoff potential [34] or the SW potential [35, 36] provides reasonable predictions for some of the nonlinear and linear properties of graphene. These two empirical potentials have fewer parameters than the Brenner potential, thus they are computationally faster than the Brenner potential. Finally, the linear part of the C-C interactions in graphene can be captured using valence force field models (VFFMs) [37], which have the most inexpensive computational costs.

The potentials of these four computation techniques can also be used for SLMoS₂. First-principles calculations can be used for SLMoS₂. In 2009, Liang et al. parametrized a bond-order potential for SLMoS₂ [38], which was based on the bond-order concept underlying the Brenner potential [33]. This Brenner-like potential was further modified to study the nanoindentations in SLMoS₂ thin films using a molecular statics approach [39]. Recently, we parametrized the SW potential for SLMoS₂, where the potential parameters were fitted to the phonon spectrum [28]. This potential could easily be used in some of the popular simulation packages, such as the General Utility Lattice Program (GULP) [40] and the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [41]. In 1975, Wakabayashi et al. [30]. developed a VFFM to calculate the phonon spectrum in bulk MoS₂. This linear model has been used to study the lattice dynamical properties of some MoS₂-based materials [42–44].

3 Phonon dispersion

A phonon is a quasiparticle in reciprocal space. Each phonon mode describes a particular type of collective vibrations for all of the atoms in the real lattice space. The symmetry of the vibration morphology follows an irreducible representation of the system space group, where the irreducible representations are denoted by the wave vector k. The phonon modes are denoted by the wave vector k and the branch index τ, where k is the inter-cell degree of freedom and τ corresponds to the intra-cell degree of freedom. Each phonon mode has a specific angular frequency ({\omega }_k^{\tau }) and eigenvector ({\xi }_k^{\tau }). For graphene and SLMoS₂, each degree of freedom in the real lattice space can be indicated by (l_1{l}_{2}s\alpha ). Here, l₁ and l₂ denote the position of the unit cell, s describes the different atoms in the unit cell, and \alpha =x,y,z is the direction of the axis. The frequency and the eigenvector of the phonon mode can be obtained through diagonalization of the following dynamical matrix:

\begin{array}{}& \begin{array}{c}{D}_{s\alpha ;s^{\prime }\beta }(k)={\displaystyle \frac{1}{\sqrt{m_s{m}_{s^{\prime }}}}}{\displaystyle \sum _{l_1=1}^{N_1}}{\displaystyle \sum _{l_2=1}^{N_2}}{K}_{00s\alpha ;l_1{l}_2{s}^{\prime }\beta }{\mathrm{e}}^{\mathrm{i}k\cdot R_{l_1{l}_2}};\hfill \\ {\displaystyle \sum _{s^{\prime }\beta }}{D}_{s\alpha ;s^{\prime }\beta }\left(k\right){\xi }_{\beta }^{({\tau }^{\prime })}(k|00s^{\prime })={\omega }^{(\tau )2}(k){\xi }_{\alpha }^{({\tau }^{\prime })}(k|00s).\hfill \end{array}\end{array}

The force constant matrix K_{00s\alpha ;l_1{l}_2{s}^{\prime }\beta } stores the information on the interactions between the two degrees of freedom, (00s\alpha) and (l_1{l}_2{s}^{\prime }\beta). The total number of unit cells is given by N_1\times N_2. For the short-range interactions, a summation over (l_1,l_2) can be truncated to the summation over the neighboring atoms.

Figure 2 shows the phonon dispersion of graphene along the high-symmetry ΓKM lines in the first Brillouin zone. The force constant matrix was constructed using the Brenner potential [33]. The inset shows the first Brillouin zone for the hexagonal lattice structure. There are six phonon branches in graphene according to the two nonequivalent carbon atoms in the unit cell. These branches (from low to high frequency) are the z-directional acoustic (ZA), transverse acoustic (TA), longitudinal acoustic (LA), z-directional optical (ZO), transverse optical (TO), and longitudinal optical (LO) branches. The three blue curves in the lower frequency range correspond to the three acoustic branches, while the upper three red curves correspond to the optical branches. The eigenvectors of the six phonon modes at the Γ point in the first Brillouin zone of graphene are displayed in Fig. 3. In the top panel, the three acoustic phonon modes have zero frequency, indicating that the interatomic potential did not vary during rigid translational motion. In the bottom panel, the two in-plane optical phonon modes have almost the same frequency, revealing the isotropic phonon properties for the two in-plane directions in graphene [45].

Figure 4 shows the phonon dispersion of SLMoS₂ along the high-symmetry ΓKM lines in the first Brillouin zone. The atomic interactions are described by the SW potential [28]. The inset shows the same first Brillouin zone as that shown for graphene. Each unit cell has one Mo atom and two S atoms; thus, there are nine branches in the phonon spectrum. The three lower blue curves correspond to the three acoustic branches, while the six upper curves correspond to the optical branches. Figure 5 shows the eigenvectors for the nine phonons at the Γ point in the first Brillouin zone of SLMoS₂. There are two interesting shear-like phonon modes and two inter-layer breathing-like phonon modes shown in the second row [46]. These interlayer phonon modes are closely related to the interlayer interaction, so they can be used to detect the interlayer coupling in heterostructures.

From the phonon dispersion of graphene and SLMoS₂, it is difficult to determine which material has better phonon properties. However, there are two obvious differences in their phonon dispersions. First, the overall spectrum of graphene is higher than that of SLMoS₂ by a factor of approximately three. As a result, the phonon modes in graphene can carry more energy than those in SLMoS₂ in the thermal transport phenomenon, leading to the stronger thermal transport ability of graphene. Second, there is a distinct energy band gap between the acoustic and optical branches in SLMoS₂. This band gap forbids many phonon–phonon scattering channels in SLMoS₂; thus, it protects the acoustic phonon modes from being interrupted by the high-frequency optical phonons [47]. As a result, SLMoS₂ nanoresonators have a higher quality (Q)-factor than graphene nanoresonators since the resonant oscillations in SLMoS₂ (related to the ZA mode) are affected by weaker thermal vibrations.

4 Mechanical properties

The mechanical properties for both graphene and SLMoS₂ have been extensively investigated [48–60]. Here, we discuss several of the basic mechanical properties, including the Young’s modulus, yield stress, bending modulus and buckling phenomenon. These mechanical properties are fundamental for the application of graphene or SLMoS₂ in nano-devices. Good mechanical stability is essential in nanoscale devices as they are sensitive to external perturbations because of their high surface-to-volume ratio.

Young’s modulus. The Young’s modulus is given by Y=E^{2D}/h, where h is the film thickness and E^{2D} is the thickness-independent effective Young’s modulus. We discuss this effective Young’s modulus here. The thicknesses were chosen to be 3.35 Å and 6.09 Å for graphene and SLMoS₂, respectively. These value are half of the lattice constant in the bulk graphite and MoS₂, respectively. Nanoindentation experiments have measured the effective Young’s modulus of graphene to be approximately 335.0 N\cdotm^-1 [3]. This value could be theoretically reproduced using a simple approach in which the nonlinear interactions are estimated from the Tersoff–Brenner potential [61].

For SLMoS₂, similar nanoindentation experiments obtained an average value for the effective Young’s modulus of 180\pm 60 N\cdotm^-1 in the experiment by Bertolazzi et al. [62], and 120\pm 30 N\cdotm^-1, measured by Cooper et al. [4, 63] Recently, Liu et al. performed similar nanoindentation experiments on chemical-vapor-deposited SLMoS₂, obtaining an effective Young’s modulus of about 170 N\cdotm^-1 [64]. The nanoindentation set up has also been used to study the Young’s modulus of thicker MoS₂ films [65]. The theoretical prediction of the effective Young’s modulus is 139.5 N\cdotm^-1 for SLMoS₂, based on the SW potential [28].

Yield stress. The nanoindentation measurements can also be used to determine the yield stress ({\sigma }_{\mathrm{int}}, the maximum of the stress-strain curve). Lee et al. determined the yield stress to be 42\pm 4 N\cdotm^-1 for graphene [3]. Moreover, the yield stresses obtained with the continuum elasticity theory were 42.4 N\cdotm^-1 using a tight-binding atomistic model [66] and 44.4 N\cdotm^-1 using the Brenner potential [67]. While the elasticity continuum simulation provided an isotropic value for the yield stress in graphene, microscopic atomic models have predicted the yield stress to be dependent on the chirality in graphene. First-principles calculations predicted the yield stress to be 40.5 N\cdotm^-1 in the zigzag direction and 36.9 N\cdotm^-1 in the armchair direction in graphene [68]. Molecular mechanics simulations obtained a yield stress of 36.9 N\cdotm^-1 in the zigzag direction and 30.2 N\cdotm^-1 in the armchair direction in graphene [68]. Both of the atomic models showed that graphene has a higher yield stress in the zigzag direction than in the armchair direction. Note that the definition of the armchair and zigzag directions in Ref. [48] opposes that in Ref. [68]. We have retained the definition from Ref. [68], where the armchair direction is along the direction of the carbon–carbon bonds.

In SLMoS₂, nanoindentation experiments found that the yield stress was 15\pm 3 N\cdotm^-1, determined by Bertolazzi et al. [62], and 16.5\pm N\cdotm^-1, determined by Cooper et al. [4, 63] In the first-principles calculations, the yield stress was predicted to be 17.5 N\cdotm^-1 under a biaxial strain in SLMoS₂ [69]. While the studies on the yield stress in SLMoS₂ are limited, the novel structure transition in SLMoS₂ has received considerable attention [70–73]. In this structure transition, the outer two S atomic layers are shifted relative to each other, leading to abrupt changes in the electronic and phonon properties in SLMoS₂. This structure transition is the result of the trilayer configuration of SLMoS₂ and is not observed in graphene.

Bending modulus. Graphene is extremely soft in the out-of-plane direction owing to its one-atom-thick structure [74–78]. Graphene is so thin that it has an extremely small bending modulus, which can be explained by the well-known relationship in the shell theorem, D=E^{2D}{h}^2/(12(1-{\nu }^2)), where h is the thickness and ν is Poisson’s ratio. The bending modulus of graphene has been derived analytically from two equivalent approaches; the obtained value was 1.17 eV using the geometric approach with interactions described by the VFFM [5, 6] and 1.4 eV from the exponential Cauchy-Born rule using the Brenner potential [7, 8]. Note that these two approaches are equivalent to each other, and the difference in the bending modulus is primarily a result of the different potentials used in these two studies.

A similar analytic approach was used to derive the bending modulus of SLMoS₂ using the SW potential [9]. The bending modulus of SLMoS₂ is 9.61 eV, which is larger than that of graphene by a factor of seven. The large bending modulus of SLMoS₂ is due to its trilayer atomic structure, which results in more interaction terms inhibiting the bending motion. The bending modulus can be calculated as follows:

\begin{array}{}\text{(1)}& D={\displaystyle \frac{{\partial }^{2}W}{\partial {\kappa }^2}},\end{array}

where W is the bending energy density and κ is the bending curvature. For SLMoS₂, the bending energy can be written as [9]

\begin{array}{}\text{(2)}& D={\displaystyle \sum _q}{\displaystyle \frac{{\partial }^{2}W}{\partial r_q^2}}{\left({\displaystyle \frac{\partial r_q}{\partial \kappa }}\right)}^2+{\displaystyle \sum _q}{\displaystyle \frac{{\partial }^{2}W}{\partial {\theta }_q^2}}{\left({\displaystyle \frac{\partial {\theta }_q}{\partial \kappa }}\right)}^2,\end{array}

where r_q and {\theta }_q are the geometrical parameters in the empirical potential expressions. This formula is substantially different from the bending modulus formula for graphene [79]. Specifically, the first derivatives, \frac{\partial r_q}{\partial \kappa } and \frac{\partial {\theta }_q}{\partial \kappa }, are nonzero owing to the trilayer structure of SLMoS₂. As a result, the bending motion in SLMoS₂ will be counteracted by an increasing number of cross-plane interactions.

Buckling phenomenon. The buckling phenomenon can be disastrous in nanoscale devices; however, it can be useful in some situations [80–83]. The Euler buckling theorem states that the buckling critical strain can be determined from the effective Young’s modulus and the bending modulus through the following formula [84]:

\begin{array}{}\text{(3)}& {ϵ}_c=-{\displaystyle \frac{4{\pi }^{2}D}{E^{2D}{L}^2}},\end{array}

where L is the length of the system. For graphene, the values of E^{2D}=335 N\cdotm^-1 and D=1.4 eV, described above, give the following explicit value for the buckling critical strain:

\begin{array}{}\text{(4)}& {ϵ}_c=-{\displaystyle \frac{2.64}{L^2}}.\end{array}

The length (L) is given in Angstroms (Å).

For SLMoS₂, the values of E^{2D}=139.5 N\cdotm^-1 and D=9.61 eV, described above, can be used to obtain the following explicit formula for the buckling critical strain:

\begin{array}{}\text{(5)}& {ϵ}_c=-{\displaystyle \frac{43.52}{L^2}}.\end{array}

For samples of the same length, the buckling critical strain for SLMoS₂ is twenty times larger than of graphene; in other words, SLMoS₂ does not easily buckle under external compression. This phenomenon has been examined with both molecular dynamics (MD) simulations and phonon analysis [71, 85].

In our discussion of the mechanical properties of graphene and SLMoS₂, we have shown that graphene has a larger Young’s modulus, larger yield stress, and is more flexibility than SLMoS₂. On the other hand, SLMoS₂ has a higher bending modulus and does not buckle as readily as graphene under external compression. Hence, in terms of the mechanical properties, it is more advantageous for graphene and SLMoS₂ to be used together in a heterostructure to mitigate the negative mechanical properties of each constituent.

5 Nanomechanical resonators

Nanoresonators based on two-dimensional materials such as graphene and SLMoS₂ are promising candidates for ultra-sensitive mass sensing and detection because of their large surface areas and small masses [86–94]. For sensing applications, it is important that the nanoresonator exhibits a high Q-factor because the sensitivity of a nanoresonator is inversely proportional to its Q-factor [95]. The Q-factor is a quantity that records the total number of oscillation cycles of the resonator before its resonant oscillation decays considerably. Hence, a weaker energy dissipation leads to a higher Q-factor.

For graphene nanoresonators, the Q-factor increases exponentially with decreasing temperatures [96, 97], T^{-\alpha }. Zande et al. [96] found that the exponent α=0.35\pm 0.05for temperatures below 40 K. For temperatures above 40 K, \alpha =2.3\pm 0.1. Chen et al. [97] observed a similar transition in the Q-factor. This continuous transition for the temperature dependence of the Q-factor is attributed to the diffusion of adsorbs in the out-of-plane direction on the surface of the graphene layer [98, 99]. The MD simulations also predicted a discontinuous transition in the Q-factor at the low temperature of 7.0 K, which is caused by the in-plane diffusion of adsorbs on the graphene surface [98]. A very high Q-factor has been achieved in the laboratory at low temperatures. Bunch et al. observed a Q-factor of 9000 for a graphene nanoresonator at 10 K [96]. Chen et al. also found that the Q-factor increased with decreasing temperature, reaching {10}^4 at 5 K [97]. Eichler et al. [100] found that the Q-factor of a graphene nanoresonator reached {10}^5 at 90 mK.

For SLMoS₂, two recent experiments demonstrated the nanomechanical resonant behavior in SLMoS₂ [101] and few-layer MoS₂ [102]. Castellanos-Gomez et al. found that the figure of merit, i.e., the frequency-Q-factor product, is f_0\times Q\approx 2\times {10}^9 Hz for SLMoS₂ [101]. Lee et al. found that few-layer MoS₂ resonators exhibit a high figure of merit of f_0\times Q\approx 2\times {10}^{10} Hz [102]. The high Q-factor of SLMoS₂ could be attributed to the energy band gap in the phonon dispersion of SLMoS₂, which protects the resonant oscillations from being scattered by thermal vibrations [47]. As a result, the Q-factor of SLMoS₂ nanoresonators was predicted to be higher than that of graphene nanoresonators by at least a factor of four.

Although it has been theoretically predicted that MoS₂ will have better mechanical resonance behavior than graphene, experiments on MoS₂ nanoresonators are limited. More measurements are needed to examine the nanoresonator properties, such as the mass sensitivity. Furthermore, the sensor application of the nanoresonators depends on the level of low frequency 1/f noise, which is a limiting factor for communication applications and sensor sensitivity as well as the selectivity of graphene and MoS₂ nanoresonators [103–109].

6 Thermal conductivity

The thermal transport phenomenon occurs when there is a temperature gradient in a material, and thermal energy can be carried by phonons or electrons. Moreover, the electronic thermal conductivity is important for metals. For graphene, phonons are the primary contributors to the thermal conductivity, and electronic thermal conductivity is less than 1% of the overall thermal conductivity [110, 111]. Thus, only the phonon (lattice) thermal conductivity will be discussed for the graphene. The thermal transport is in the ballistic regime at low temperatures with weak phonon–phonon scattering [112, 113]. For ballistic transport, the thermal conductivity (κ) is proportional to the length (L) of the system: \kappa \propto L. At high temperatures, the phonon-phonon scattering is strong, so the thermal transport is in the diffusive regime. For diffusive transport, the thermal conductivity is related to the thermal current density (J) and the temperature gradient (\nabla T) through the Fourier law: \kappa =-\nabla T/J.

In bulk materials, the room temperature thermal conductivity is normally a size-independent constant. Graphene has high thermal conductivity [49, 114–124], which behaves irregularity with the length in Q2D graphene; in other words, the in-plane thermal conductivity is not constant and increases with increasing sample length [125–133]. For 10 nm long graphene, the room-temperature thermal conductivity from the MD simulation is on the order of 60 W\cdotm^-1\cdotK^-1 [134]. The thermal conductivity increased quickly with increasing length, reaching 250 W\cdotm^-1\cdotK^-1 for 300 nm long graphene [129]. For a length of 4.0 μm, graphene had a thermal conductivity near [135] 2500 W\cdotm^-1\cdotK^-1. For graphene samples larger than 10 μm, the thermal conductivity varies in the range from 1500 W\cdotm^-1\cdotK^-1 to 5000 W\cdotm^-1\cdotK^-1, depending on the sample size and quality [129, 136, 137]. Balandin et al. reported the highest value of 5000 W\cdotm^-1\cdotK^-1 for a 20 μm sample. Thus, these studies show that the thermal conductivity in graphene increases with increasing dimension, even though the sample size is larger than the phonon mean free path [137]. However, there is no universally accepted underlying mechanism for the size dependence of the thermal conductivity in graphene [127, 132, 138–141]. In the out-of-plane direction, the thermal conductivity for graphene decreases with increasing layer number as there are more phonon-phonon scattering channels in thicker few-layer graphene [138, 142–149].

For MoS₂, a recent experiment by Sahoo et al. found that few-layer MoS₂ has a thermal conductivity near [150] 52 W\cdotm^-1\cdotK^-1, which is much lower than that in thick graphene layers (1000 W\cdotm^-1\cdotK^-1) [142]. Although there are currently no measurements on the thermal conductivity of SLMoS₂, this topic has attracted increasing interest in the theoretical community [151–153]. In 2010, Varshney et al. performed force-field-based MD simulations to study the thermal transport in SLMoS₂ [151]. In 2013, two first-principles calculations were performed to investigate the thermal transport in SLMoS₂ in the ballistic transport regime [152, 153]. The predicted room-temperature thermal conductivity in the ballistic regime was below 800 W\cdotm^-1\cdotK^-1 for a 1.0 μm long SLMoS₂ sample [153]. This value is considerably lower than the ballistic thermal conductivity of 5000 W\cdotm^-1\cdotK^-1 for a graphene sample with the same length [154]. The smaller thermal conductivity of SLMoS₂ in the ballistic regime was caused by the lower phonon spectrum in SLMoS₂, which is lower than that of graphene by a factor of three. Thus, each phonon mode in SLMoS₂ carries less thermal energy than that in graphene. In 2013, we performed MD simulations to predict the room-temperature thermal conductivity of SLMoS₂, which was 6.0 W\cdotm^-1\cdotK^-1 for a 4.0 nm long system [28]. More recently, the size dependence of the thermal conductivity in SLMoS₂ was studied with MD simulations. The value obtained was below 2.0 W\cdotm^-1\cdotK^-1 for a system with a length shorter than 120.0 nm [155].

As we have shown, graphene has a much higher thermal conductivity than SLMoS₂. The high thermal conductivity of graphene is useful for transporting heat out of electronic transistor devices; thus, graphene can be used to enhance the thermal transport capability of some composites [156–164]. Current transistors operate at very high speeds and are damaged by the inevitable Joule heating if the generated heat energy is not pumped out effectively. In this sense, graphene has better thermal conductivity properties than SLMoS₂.

7 Electronic band structure

The electronic band structure is fundamental for electronic processes such as transistor performance. In particular, the value of the electronic band gap determines whether the material is a conductor (zero band gap), semiconductor (moderate band gap), or insulator (large band gap).

Electrons in graphene behavior like Dirac fermions; i.e., the electronic energy dispersion is linear near the Brillouin zone corner. The velocity of this Dirac fermion is at 1/300 the speed of light [165–167]. The Dirac fermion was found to be closely related to the mirror plane symmetry in AB-stacked few-layer graphene; i.e., Dirac fermions exist in AB-stacked few-layer graphene with an odd number of layers, and the electronic spectrum becomes parabolic in AB-stacked few-layer graphene with an even number of layers [168]. Interestingly, Dirac fermions are also present in twisted bilayer graphene [169] due to the effective decoupling of the two graphene layers by the twisting defect; that is, the mirror plane symmetry is effectively recovered in twisted bilayer graphene. The Dirac cone at the Brillouin zone corner has a zero band gap in graphene, which is mainly contributed by free π electrons [170]. For electronic devices, like transistors, a finite band gap is desirable, and various techniques have been invented to open an electronic band gap in graphene. Strain engineering can be used to generate a finite band gap of 0.1 eV with a 24% uniaxial strain [171]. Guinea et al. applied triangular symmetric strains to generate a band gap over 0.1 eV, which is observable at room temperature [172]. A finite band gap can also be opened by confining the graphene structure in a nanoribbon form, where the band gap increases with decreasing ribbon width [173].

Electrons in SLMoS₂ are normal fermions with parabolic energy dispersion, and SLMoS₂ is a semiconductor with a direct band gap above 1.8 eV [174–177]. This finite band gap endorses SLMoS₂ for transistor applications [178, 179]. Similar to graphene, the band gap in SLMoS₂ can also be modulated by strain engineering. First-principles calculations predict a semiconductor-to-metal transition in SLMoS₂ using biaxial compression or tension [180]. An experiment by Eknapakul et al. shows that a uniaxial tensile mechanical strain of 1.5% can produce a direct-to-indirect band gap transition [181]. With an increasing number of layers, the electronic band gap for few-layer MoS₂ undergoes a direct-to-indirect transition, decreasing to a value of 1.2 eV for bulk MoS₂ [182].

From these comparisons, we find that SLMoS₂ possesses a finite band gap prior to any gap-opening engineering. Consequently, SLMoS₂ may be more competitive than graphene for applications in transistors, optoelectronics, energy harvesting, and other nano-material fields.

8 Optical absorption

The optical properties of Q2D materials are important for their applications in photodetectors, phototransitors, or other photonic nanodevices. Moreover, the photocarriers in these Q2D materials may have significantly different behavior from conventional semiconductors due to their particular configurations.

Graphene has a Dirac cone electron band structure with zero band gap [166, 167]. Related to this unique band structure, graphene can absorb about 2% of the incident light over a broad wavelength, which is strong considering its single-layer structure [183]. Xia et al. demonstrated an ultra-fast photodetector behavior for graphene, where the photoresponse did not degrade for optical intensity modulations up to 40 GHz and where the intrinsic band width was estimated to be above 500 GHz [184]. However, the photoresponsivity for graphene is low due to its zero bandgap.

On the other hand, SLMoS₂ has a direct band gap of about 1.8 eV [174, 175]. This optical-range band gap leads to a high absorption coefficient for incident light, so SLMoS₂ has very high sensitivity in photon detection [182]. Lopez-Sanchez et al. found that the photoresponsivity of SLMoS₂ can be as high as 880 A\cdotW^-1 for incident light at the wavelength of 561 nm, and the photoresponse is in the 400–680 nm range [185]. This high photoresponsivity together with fast light emission enables SLMoS₂ to be used as ultra sensitive phototransistors with good device mobility and large ON current. In phototransistors, the electron-hole pair can be efficiently generated by photoexcitions in doped SLMoS₂, which join the doping-induced charges to form bound states of two electrons and one hole. As a result, the carrier effective mass is considerably increased, and the photoconductivity can be decreased [186].

Regarding the optical properties of graphene and SLMoS₂, graphene shows very fast photo detection, while SLMoS₂ has very sensitive photo detection. Considering these complementary property, heterostructures designed with these two materials may be fruitful.

9 Graphene/MoS₂ heterostructure

Thus far, we have compared several of the properties of graphene and SLMoS₂. This section is devoted to studies on the close relationship between these two materials. As graphene and SLMoS₂ have complementary physical properties, it is natural to combine graphene and SLMoS₂ in specific ways to create heterostructures that mitigate any negative properties [25, 26, 187–197].

A few experiments have investigated the advanced properties of graphene/MoS₂ heterostructures. Britnell et al. found that graphene/MoS₂ heterostructures have high quality photon absorption and electron-hole creation properties because of the enhanced light-matter interactions in the SLMoS₂ layer [25]. As discussed earlier, graphene has outstanding mechanical properties. These mechanical properties have been used to protect MoS₂ films from radiation damage [26]. Recently, Larentis et al. measured the electron transport in graphene/MoS₂ heterostructures and observed a negative compressibility in the MoS₂ component [191]. This surprising phenomenon could be explained based on the interplay between the Dirac and parabolic bands for graphene and MoS₂, respectively. Yu et al. fabricated high-performance electronic circuits based on a graphene/MoS₂ heterostructure with MoS₂ as the transistor channel and graphene as the contact electrodes and the circuit interconnects [198].

Although experimentalists have shown great interest in graphene/MoS₂ heterostructures, the corresponding theoretical efforts have been limited until recently. Two first-principle studies have predicted the inter-layer space and binding energy for the heterostructure at –21.0 meV and 3.66 Å, respectively [199], and at –23.0 meV and 3.32 Å, respectively [197]. Using these two quantities [200], a set of Lennard-Jones potential parameters were determined as ϵ=3.95 meV and \sigma =3.625 Å, with a cutoff of 10.0 Å. These potential parameters were used to study the tension-induced structure transition of the graphene/MoS₂/graphene heterostructure. Moreover, the Young’s modulus of the graphene/MoS₂/graphene heterostructure could be predicted by the following rule of mixtures, based on the arithmetic average [201]:

\begin{array}{}\text{(6)}& Y_{GMG}=Y_G{f}_G+Y_M{f}_M,\end{array}

where Y_{GMG}, Y_G and Y_M are the Young’s moduli of the heterostructure, graphene, and SLMoS₂, respectively. Furthermore, f_G=2V_G/(2V_G+V_M)=0.524 is the volume fraction for the two outer graphene layers in the heterostructure, and f_M=V_M/(2V_G+V_M)=0.476 is the volume fraction for the inner SLMoS₂ layer. The thicknesses were 3.35 Å and 6.09 Å for single-layer graphene and SLMoS₂, respectively. At room temperature, Young’s modulus was 859.69 GPa for graphene and 128.75 GPa for SLMoS₂. From this mixing rule, the upper-bound of the Young’s modulus for the heterostructure was 511.76 GPa.

The total strain of the graphene/MoS₂/graphene heterostructure was about 0.26, which is much smaller than the value of 0.40 for SLMoS₂ [200]. Under large mechanical tension, the heterostructure collapsed from the buckling of the outer graphene layers. When the heterostructure was stretched in the longitudinal direction, these graphene layers were compressed in the lateral direction by Poisson-induced stress.

10 Conclusions

We have compared the mechanical properties of graphene and SLMoS₂. The primary results of this comparison are tabulated in Table 1. Moreover, Table 1 serves as a resource for predicting the corresponding properties of graphene/MoS₂ heterostructures.

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