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In-plane and cross-plane thermal conductivities of molybdenum disulfide
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 Nanotechnology 26 065703 (http://iopscience.iop.org/0957-4484/26/6/065703) View the table of contents for this issue, or go to the journal homepage for more
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Nanotechnology Nanotechnology 26 (2015) 065703 (9pp)
doi:10.1088/0957-4484/26/6/065703
In-plane and cross-plane thermal conductivities of molybdenum disulfide Zhiwei Ding1, Jin-Wu Jiang2, Qing-Xiang Pei1 and Yong-Wei Zhang1 1
Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, People’s Republic of China
2
E-mail: [email protected] and [email protected] Received 25 November 2014, revised 19 December 2014 Accepted for publication 22 December 2014 Published 19 January 2015 Abstract
We investigate the in-plane and cross-plane thermal conductivities of molybdenum disulfide (MoS2) using non-equilibrium molecular dynamics simulations. We find that the in-plane thermal conductivity of monolayer MoS2 is about 19.76 W mK−1. Interestingly, the in-plane thermal conductivity of multilayer MoS2 is insensitive to the number of layers, which is in strong contrast to the in-plane thermal conductivity of graphene where the interlayer interaction strongly affects the in-plane thermal conductivity. This layer number insensitivity is attributable to the finite energy gap in the phonon spectrum of MoS2, which makes the phonon–phonon scattering channel almost unchanged with increasing layer number. For the cross-plane thermal transport, we find that the cross-plane thermal conductivity of multilayer MoS2 can be effectively tuned by applying cross-plane strain. More specifically, a 10% cross-plane compressive strain can enhance the thermal conductivity by a factor of 10, while a 5% cross-plane tensile strain can reduce the thermal conductivity by 90%. Our findings are important for thermal management in MoS2 based nanodevices and for thermoelectric applications of MoS2. Keywords: molybdenum disulfide, thermal conductivity, molecular dynamics (Some figures may appear in colour only in the online journal) 1. Introduction
27]. A very high in-plane thermal conductivity has been found experimentally [22] and explained theoretically [13, 14]. Hence graphene is useful in the enhancement of the thermal transport capability of some composite materials [28]. Moreover, a dimensional cross-over of thermal transport from two-dimensional graphene to three-dimensional graphite has been observed in experiments, which showed that the thermal conductivity decreases monotonically with increasing layer number [17, 22]. However, for MoS2, most existing studies only focused on the in-plane thermal conductivity of monolayer [29–32]. The effect of interlayer interaction on the inplane thermal conductivity of multilayer MoS2 is still lacking. Besides, the cross-plane thermal conductivity of multi-layer MoS2 also remains unknown. In this work, we investigate both the in-plane and crossplane thermal conductivity of MoS2 using molecular dynamics (MD) simulations. It is found that the in-plane thermal conductivity of multilayer MoS2 is insensitive to the number of layers, which can be understood based on the
Monolayer and multilayer molybdenum disulfide (MoS2) has attracted great attention recently due to its unique electronic [1, 2], optical and mechanical [3–5] properties. Compared with graphene, MoS2 exhibits a direct sizable band gap [6], which makes MoS2 a promising candidate material in many electronic devices [7–10]. In addition to the potential application in field effect transistor, MoS2 has great potential to be used in thermoelectric energy conversion devices due to its high Seebeck coefficient [11]. In general, a high thermal conductivity is favorable in heat dissipation devices to spread heat while a low thermal conductivity is desirable in thermoelectric devices to enhance the thermoelectric conversion efficiency. Therefore, an in-depth understanding of the thermal conductivity is crucial in optimizing the design of MoS2 based nano-devices. The thermal conductivities of single-layer graphene and few-layer graphene have been investigated intensively [12– 0957-4484/15/065703+09$33.00
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© 2015 IOP Publishing Ltd Printed in the UK
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Table 1. Values of parameters used in L–J potential.
Atom 1
Atom 2
r0 (Å)
σ (Å)
ε (eV)
S Mo S
S Mo Mo
4.035 3.052 3.544
3.595 2.719 3.157
0.011 87 0.002 43 0.024 89
phonon spectrum and the configuration of MoS2. In addition, the effects of strain and temperature on the cross-plane thermal conductivity of multilayer MoS2 are also investigated. It is shown that strain engineering is an effective approach to tune the cross-plane thermal conductivity. For example, a 10% cross-plane compressive strain can enhance the thermal conductive by a factor of 10, while a 5% tensile cross-plane strain can reduce the thermal conductivity by 90%. Our findings here may be useful for the manipulation of the thermal transport in MoS2 and for the practical design of MoS2 based nano-devices.
Figure 1. Illustration of atomic configuration of multilayer MoS2, The atoms in each layer are assigned a number to define the van der Waals interaction in L–J potential. Yellow balls are S atoms and purple balls are Mo atoms.
2. Simulation method All MD simulations in our study are performed using LAMMPS package [33]. Stillinger–Weber (SW) potential [34] is used to describe the interatomic interaction within each layer of MoS2 while L–J potential is used to model the van der Waals interaction among different layers in multilayer MoS2. The SW potential treats the bond bending by a twobody interaction, while the angle bending is described by a three-body interaction [34]. The L–J potential has the following form ⎡ ⎛ ⎞12 ⎛ ⎞6⎤ σ σ V ( rij ) = 4ε ⎢ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥ , ⎢ ⎝ rij ⎠ r ⎝ ij ⎠ ⎥⎦ ⎣
configuration is equilibrated at 300 K by using constant pressure and constant temperature (NPT) MD simulation for 105 time steps. Then the system is switched to constant volume and constant energy (NVE) ensemble to keep the energy conserved and heat fluxes are imposed on the system by continuously adding a small amount of heat dq to the hot region and removing it from the cold region as shown in figure 2. The heat flux J can be calculated by J=
(1)
dq , 2Adt
(2)
where J is the heat flux, A is the cross-sectional area and dt is the time step. The factor of 2 is due to the fact that the heat current propagates in two opposite directions from the hot region to the cold region. The sample is divided into N slabs to evaluate the temperature gradient. The temperature T in each slab is calculated as the average temperature of the all atoms in the slab
where rij is the distance between atoms i and j. Values of σ and ε for the L–J potential are listed in table 1 [35]. In the table, r0 is the distance at which the potential reaches its minimum. We assumed and verified that the interatomic interactions in the L–J potential between atoms beyond two layers are so weak that they contribute little to the total thermal conductivity. This is because their distance is larger than 12 Å which is the cut-off distance for the L–J potential. Thus, we assign a number to each line of atoms as shown in figure 1, which provides an easy way to define the L–J potential in MD simulations for multilayer MoS2. The multilayer MoS2 is stacked in A–A′ sequence, which is the most stable stacking order with the lowest energy [36], where Mo(S) atoms of the second layer are right above S(Mo) atoms of the first layer. The non-equilibrium molecular dynamics (NEMD) simulations are performed to calculate the thermal conductivity. The schematic model setup is shown in figure 2. The equations of atomic motion are integrated with a time step of 1.0 fs. Before applying heat flux, the initial relaxed
N
T=
1 ∑mi vi2 , 3Nk i = 1
(3)
where N is the total number of atoms in the slab, mi is the mass of atom i, vi is the velocity of the atom i and k is the Boltzmann constant. We run 106 time steps to reach steady state and another 4 × 106 time steps to obtain the time-average temperature profile, which is used to calculate the temperature gradient ∂T ∂l, where l is the distance along the heat transfer direction. The thermal conductivity λ can be calculated as λ=
2
J dq = . ∂T ∂l 2Adt (∂T ∂l)
(4)
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Figure 2. Schematic setup of the non-equilibrium molecular dynamics simulation. The width of the hot region is twice of each cold region. At
each time step, a small amount of heat dq is added to the hot region and removed from cold regions to ensure the conservation of energy.
3. Results and discussion 3.1. In-plane thermal conductivity
Here, we focus on the in-plane thermal conductivity in the armchair direction (see figure 2). Periodic boundary conditions are applied in the two in-plane directions. Firstly, we study the thermal conductivity of monolayer MoS2. It is well-known that thermal conductivity predicted by NEMD usually shows strong size effect due to the phonon scattering at sample ends, especially when the sample size is below the mean-free path (MFP) [37, 38]. To obtain the thermal conductivity at infinite sample length, an inverse fitting procedure [31, 37, 39, 40] is employed. We choose the monolayer MoS2 samples with a fixed width of 10 nm and vary the length from 40 to 100 nm. Consequently, the corresponding change of the distance between the hot and cold region is from 20 to 50 nm. As shown in figure 3, the thermal conductivity at infinite length λ∞ can be obtained by linear fitting the computed thermal conductivity at different sample sizes to the following equation [39] 1 1 ⎛ L⎞ ⎜1 + ⎟, = d⎠ λ λ∞ ⎝
Figure 3. The inverse of thermal conductivity for monolayer MoS2
as a function of the inverse of the distance d between the hot and cold region. The thermal conductivity at infinite sample length can be obtained by linear extrapolating to 1/d = 0.
(5)
where λ∞ is the converged thermal conductivity at infinite length, L is the MFP, d is the distance between hot and cold region and λ is the size-dependent thermal conductivity. From the interception of the fitting line at the vertical axis, the λ∞ of the monolayer MoS2 sheet is found to be about 19.76 W mK−1, which is comparable with the previous calculated value of 23.2 W mK−1 by density functional perturbation theory [29] and close to the measured value of 34.5 W mK−1 [41]. We then proceed to study the layer effect on the thermal conductivity. We generated the MoS2 samples with the same in-plane dimensions of 20 × 20 nm2, but with different number of layers. The calculated thermal conductivity as a function of the number of layers is shown in figure 4. It can be observed that the thermal conductivity almost remains constant when the number of layers increases. This is consistent
Figure 4. In-plane thermal conductivity of MoS2 as a function of the number of layers. The thermal conductivity remains almost unchanged as the number of layers increases.
with a recent experiment, which found that the thermal conductivity of four-layer and seven-layer MoS2 are almost the same at various temperatures [30]. However, this effect is quite different from that of few-layer graphene. Experiment has shown that the thermal conductivity of few-layer graphene decreases monotonically with the increase of layer number [22]. This experiment result has also been confirmed by theoretical studies [13, 22, 25]. The reduction in thermal 3
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Figure 6. Cross-plane thermal conductivity of MoS2 as a function of temperature.
case of MoS2, there are some chemical bonds along the outof-plane direction. These out-of-plane bonds have similar chemical properties as the in-plane bonds, making multilayer MoS2 somehow more isotropic in all the directions. As a result, the peaks are similar for both in-plane and out-of-plane DOS because the vibration modes are similar in these two directions as shown in figure 5(b). Figure 5. Vibrational DOS of (a) monolayer graphene and (b)
3.2. Cross-plane thermal conductivity
monolayer MoS2.
We move on to study the cross-plane thermal conductivity of MoS2. In the current study, we focus on the variation of the thermal conductivity and thermal resistance with respect to strain and temperature. The thermal resistance R is defined as:
conductivity in few-layer graphene was attributed to the increased phonon–phonon scattering channels, which are increased by increasing layer number due to the cross-over of the acoustic and optical phonon branches in the phonon spectrum of graphene. In other words, this effect is directly related to the specific phonon spectrum of graphene, which has no energy band gap between the acoustic and optical branches. However it has been shown that the phonon spectrum in MoS2 is characterized with a finite energy gap between the acoustic and optical branches [42]. As a result, the phonon–phonon scattering channels in MoS2 are almost unchanged by increasing layer number, resulting in a layerindependent thermal conductivity. In addition, for graphene, it is assumed that there is no scattering from the top and bottom surfaces since it is only one atomic layer thick, which is too thin to have any cross-plane velocity component [14, 22]. Therefore, the surface roughness parameter p in Ziman formula was set to 1 for the top and bottom surfaces in previous studies [14, 22]. However, due to the intrinsic trilayer structure in each MoS2 layer, the surface roughness of the top and bottom surfaces is important even for monolayer MoS2. As a result, the thermal conductivity in monolayer MoS2 is close to that in multilayer MoS2. This point could be illustrated by the difference in the phonon power spectrum of graphene and MoS2. The vibrational density of states (DOS) are calculated by taking the Fourier transform of the velocity auto-correlation function and the AIREBO potential [43] is adopted for the interatomic interaction in graphene. For graphene, the peak of the total DOS is almost consistent with the peak of in-plain DOS and the shapes of total DOS and inplane DOS are quite close, as shown in figure 5(a). For the
R = D/ λ ,
(6)
where D is the interlayer distance. In the present simulation, the interlayer distance D is chosen as 0.621 nm [36]. 3.2.1. Effect of temperature. Although the in-plane thermal
conductivities of MoS2 at different temperatures have been investigated [34, 44], the effect of temperature on the crossplane thermal conductivity has not been reported. Here we varied the temperature from 100 to 500 K, and the simulated cross-plane thermal conductivity of MoS2 is shown in figure 6. It can be seen that the cross-plane thermal conductivity decreases as the temperature increases from 100 to 500 K. It can also be seen that the temperature-induced reduction is quite significant: the thermal conductivity at 500 K is only about 30% of that at 100 K. 3.2.2. Effect of cross-plane strain. To study the strain effect
on the cross-plane heat transport in MoS2, we choose a model of 30-layer structure with a cross-sectional area 4 × 4 nm2 as shown in figure 7. After the model is fully relaxed, a crossplane strain is applied on the equilibrium structure. The strain is defined as ε = ( l − lo )/ lo,
(7)
where lo and l are the length of the simulation box in the cross-plane direction for the unstrained and strained MoS2 sample, respectively. Note that the strain is applied by 4
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Figure 7. Simulation model of 30-layer MoS2 for studying the cross-plane thermal conductivity. The orientation is indicated by the coordinate. The heat flux is applied in the cross-plane (z) direction.
Figure 8. Mechanical response during the straining process. (a) and (b) stress components induced in all directions. (c) and (d) standard
deviation of the z-coordinate of S and Mo atoms. (e) and (f) distance between S atoms in the same MoS2 layer (SS) and in the two adjacent layers (SS′).
deforming the simulation box size in the cross-plane direction at a constant strain rate of 5 × 108 s−1 while keeping the box size along the other two directions unchanged (NVT ensemble). Figure 8 presents the mechanical response of multilayer MoS2 upon the application of the strain. When compressive strain εc is applied, the stress components, Sxx, Syy, Szz, increase monotonically, while a much bigger stress is induced in the z direction and the stresses in the x and y direction are almost the same (figure 8(a)). It should be noted that the presence of the in-plane stress is due to the trilayer structure of MoS2, while no in-plane stress is generated for graphene since graphene only consists of one atom layer [45]. When tensile strain εc is applied, the stress components increase monotonically till the strain reaches around 0.07, at which the
tensile strain overcomes the L–J potential, the system breaks and the stress components return near zero, as shown in figure 8(d). Moreover the standard deviations of the zcoordinate of S atom (S_S) and Mo atom (S_Mo) remain at very low values during compressing or stretching process, as shown in figures 8(b) and (e), which indicate that no buckling occurs. In addition, the variations of the distance between S atoms in the same layer (SS) as well as in two adjacent layers (SS′) (see figure 9) are also calculated and shown in figures 8(c) and (f). It can be seen that the SS remains almost unchanged as compared with SS′, especially during the stretching process. This is because the bonding interaction between atoms in the same MoS2 layer (SW potential) is much stronger than the non-bonding van der Waals interaction (L–J potential). It is noted that there is some difference in 5
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Figure 9. Schematic illustration to the definition of distance between S atoms in the same MoS2 layer SS and in the two adjacent layers SS′.
SS′ between figures 8(c) and (f). This difference in SS′ during the compressing and stretching process is caused by the asymmetry of the L–J potential. The cross-plane thermal conductivity and resistance at room temperature under various cross-plane strains are shown in figure 10. The calculated values are normalized by the corresponding values at zero strain. It is seen that with a 10% compressive strain, which corresponds to a compressive stress of about 14 GPa, is able to enhance the thermal conductivity more than 14 times and reduce the thermal resistance by 94%. A 5% tensile strain, which induces about 1.5 GPa tensile stress, is able to reduce the thermal conductivity by nearly 90% and increase the thermal resistance by a factor of 10. It is noted that this trend is in good agreement with the simulation result of bulk argon using L–J potential [46, 47]. To understand this trend, the vibrational DOS are calculated as shown in figure 11. From the kinetic theory, the thermal conductivity λ can be expressed as λ=
∑Ci vi L i,
Figure 11. Vibrational DOS of multilayer MoS2 under different cross-plane strains. (a) Out-of-plane vibrational DOS. (b) In-plane vibrational DOS. (c) Total vibrational DOS.
The enhanced force constant increases the phonon frequency, thus causing the blue shift of the major peaks in the phonon spectrum as shown in figure 11. Such a blue shift results in an increase in both group velocity and specific heat [48, 49], as a result, the thermal conduction increases according to equation (8). Similarly, tensile strain softens the phonon
(8)
i
where Ci, vi, Li are, respectively, the specific heat, group velocity and MFP of phonon mode i. When compressive strain is applied, the interlayer L–J interaction gets enhanced.
Figure 10. (a) Cross-plane thermal conductivity and (b) thermal resistance as a function of cross-plane strain. 6
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Figure 12. Mechanical response during the straining process. (a) and (b) stress components induced in all direction. (c) and (d) standard deviation of the z-coordinate of S atom and Mo atoms. (e) and (f) distance between S atoms in the same MoS2 layer SS and in two adjacent layers SS′.
Figure 13. Variations of (a) cross-plane thermal conductivity and (b) cross-plane thermal resistance as a function of biaxial in-plane strain.
standard deviations of the z-coordinate of S atoms (S_S) and Mo atoms (S_Mo). It was shown that the thermal transport behavior of buckled structure is quite different from that of flat ones [49], hence, it is important to check whether buckling occurs within the strain range. We find that the values of S_S and S_Mo in figures 12(b) and (e) remain at very low values during compressing, thus buckling does not occur within the compression strain range. We also double checked the atomic configuration of the compressed structure at 10% compressive strain and no buckling was observed. The variations of the cross-plane thermal conductivity and resistance as a function of the in-plane strain are shown in figure 13. The tensile strain results in an increase of thermal resistance and a decrease of thermal conductivity, while the compressive strain leads to a decrease of thermal resistance and an increase of thermal conductivity. This effect of inplane strain on the cross-plane thermal conductivity and
vibration and results in a red shift of the major peaks, which reduces the thermal conductivity. In addition, the inter-layer distance d decreases during the compressing process and increases during the stretching process, thus resistance R decreases when compressive strain is applied and increases when tensile strain is applied as shown in figure 10(b). 3.2.3. Effect of in-plane strain. We then move on to study the
effect of in-plane strain on the cross-plane thermal transport. Biaxial strain is applied in the x and y directions of the simulation model (the in-plane of MoS2), while the model size in the z direction is fixed. When biaxial compressive strain is applied, compressive stresses in all directions increase significantly, see figure 12(a). When tensile strain is applied, tensile stresses in the x and y directions increase significantly while the stress in the z direction shows a much smaller increase, see figure 12(d). We also calculated the 7
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4. Conclusion We studied the in-plane and cross-plane thermal conductivities of MoS2 by using NEMD. A value of around 19.76 W mK−1 for the in-plane thermal conductivity is obtained. We further found that the in-plane thermal conductivity of multilayer MoS2 is insensitive to the number of layers, which may be explained by the finite energy gap between the acoustic and optical branches in the phonon spectrum and its intrinsic trilayer structure. The cross-plane thermal conductivity is found to decrease when the temperature rises from 100 to 500 K. Moreover, we also studied strain effect on the cross-plane thermal conductivity and found that the cross-plane strain has a much stronger effect than the in-plane strain. A 10% cross-plane compressive strain can enhance the thermal conductivity by a factor of 10, while a 5% cross-plane tensile strain can reduce the thermal conductivity by 90%. Our findings may be useful for thermal management in MoS2 based nanodevices and for thermoelectric applications of MoS2.
Acknowledgments The work is supported by the Agency for Science, Technology and Research (A*STAR), Singapore and by the Recruitment Program of Global Youth Experts of China and start-up funding from the Shanghai University.
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resistance is similar to that of cross-plane strain as shown in figure 10 in the previous section. Comparing figures 10 and 13, we observe that the cross-plane strain has a stronger effect on the thermal conductivity and resistance than the in-plane strain, indicating that the interlayer thermal resistance plays a major role in the whole cross-plane thermal resistance. Compared to the in-plane strain, the cross-plane strain is more effective in tuning the cross-plane thermal conductivity of MoS2. The variations of cross-plane thermal conductivity and resistance can also be understood from the phonon spectra. As shown in figure 14, compressive/tensile strain can induce a blue/red shift of the phonon spectrum and result in an increase/decrease in the group velocity and specific heat [48, 49], thus leading to an increase/decrease of cross-plane thermal conductivity and decrease/increase of cross-plane thermal resistance. 8
Nanotechnology 26 (2015) 065703
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In-plane and cross-plane thermal conductivities of molybdenum disulfide
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Nanotechnology Nanotechnology 26 (2015) 065703 (9pp)
doi:10.1088/0957-4484/26/6/065703
In-plane and cross-plane thermal conductivities of molybdenum disulfide Zhiwei Ding1, Jin-Wu Jiang2, Qing-Xiang Pei1 and Yong-Wei Zhang1 1
Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, People’s Republic of China
2
E-mail: [email protected] and [email protected] Received 25 November 2014, revised 19 December 2014 Accepted for publication 22 December 2014 Published 19 January 2015 Abstract
We investigate the in-plane and cross-plane thermal conductivities of molybdenum disulfide (MoS2) using non-equilibrium molecular dynamics simulations. We find that the in-plane thermal conductivity of monolayer MoS2 is about 19.76 W mK−1. Interestingly, the in-plane thermal conductivity of multilayer MoS2 is insensitive to the number of layers, which is in strong contrast to the in-plane thermal conductivity of graphene where the interlayer interaction strongly affects the in-plane thermal conductivity. This layer number insensitivity is attributable to the finite energy gap in the phonon spectrum of MoS2, which makes the phonon–phonon scattering channel almost unchanged with increasing layer number. For the cross-plane thermal transport, we find that the cross-plane thermal conductivity of multilayer MoS2 can be effectively tuned by applying cross-plane strain. More specifically, a 10% cross-plane compressive strain can enhance the thermal conductivity by a factor of 10, while a 5% cross-plane tensile strain can reduce the thermal conductivity by 90%. Our findings are important for thermal management in MoS2 based nanodevices and for thermoelectric applications of MoS2. Keywords: molybdenum disulfide, thermal conductivity, molecular dynamics (Some figures may appear in colour only in the online journal) 1. Introduction
27]. A very high in-plane thermal conductivity has been found experimentally [22] and explained theoretically [13, 14]. Hence graphene is useful in the enhancement of the thermal transport capability of some composite materials [28]. Moreover, a dimensional cross-over of thermal transport from two-dimensional graphene to three-dimensional graphite has been observed in experiments, which showed that the thermal conductivity decreases monotonically with increasing layer number [17, 22]. However, for MoS2, most existing studies only focused on the in-plane thermal conductivity of monolayer [29–32]. The effect of interlayer interaction on the inplane thermal conductivity of multilayer MoS2 is still lacking. Besides, the cross-plane thermal conductivity of multi-layer MoS2 also remains unknown. In this work, we investigate both the in-plane and crossplane thermal conductivity of MoS2 using molecular dynamics (MD) simulations. It is found that the in-plane thermal conductivity of multilayer MoS2 is insensitive to the number of layers, which can be understood based on the
Monolayer and multilayer molybdenum disulfide (MoS2) has attracted great attention recently due to its unique electronic [1, 2], optical and mechanical [3–5] properties. Compared with graphene, MoS2 exhibits a direct sizable band gap [6], which makes MoS2 a promising candidate material in many electronic devices [7–10]. In addition to the potential application in field effect transistor, MoS2 has great potential to be used in thermoelectric energy conversion devices due to its high Seebeck coefficient [11]. In general, a high thermal conductivity is favorable in heat dissipation devices to spread heat while a low thermal conductivity is desirable in thermoelectric devices to enhance the thermoelectric conversion efficiency. Therefore, an in-depth understanding of the thermal conductivity is crucial in optimizing the design of MoS2 based nano-devices. The thermal conductivities of single-layer graphene and few-layer graphene have been investigated intensively [12– 0957-4484/15/065703+09$33.00
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© 2015 IOP Publishing Ltd Printed in the UK
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Table 1. Values of parameters used in L–J potential.
Atom 1
Atom 2
r0 (Å)
σ (Å)
ε (eV)
S Mo S
S Mo Mo
4.035 3.052 3.544
3.595 2.719 3.157
0.011 87 0.002 43 0.024 89
phonon spectrum and the configuration of MoS2. In addition, the effects of strain and temperature on the cross-plane thermal conductivity of multilayer MoS2 are also investigated. It is shown that strain engineering is an effective approach to tune the cross-plane thermal conductivity. For example, a 10% cross-plane compressive strain can enhance the thermal conductive by a factor of 10, while a 5% tensile cross-plane strain can reduce the thermal conductivity by 90%. Our findings here may be useful for the manipulation of the thermal transport in MoS2 and for the practical design of MoS2 based nano-devices.
Figure 1. Illustration of atomic configuration of multilayer MoS2, The atoms in each layer are assigned a number to define the van der Waals interaction in L–J potential. Yellow balls are S atoms and purple balls are Mo atoms.
2. Simulation method All MD simulations in our study are performed using LAMMPS package [33]. Stillinger–Weber (SW) potential [34] is used to describe the interatomic interaction within each layer of MoS2 while L–J potential is used to model the van der Waals interaction among different layers in multilayer MoS2. The SW potential treats the bond bending by a twobody interaction, while the angle bending is described by a three-body interaction [34]. The L–J potential has the following form ⎡ ⎛ ⎞12 ⎛ ⎞6⎤ σ σ V ( rij ) = 4ε ⎢ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥ , ⎢ ⎝ rij ⎠ r ⎝ ij ⎠ ⎥⎦ ⎣
configuration is equilibrated at 300 K by using constant pressure and constant temperature (NPT) MD simulation for 105 time steps. Then the system is switched to constant volume and constant energy (NVE) ensemble to keep the energy conserved and heat fluxes are imposed on the system by continuously adding a small amount of heat dq to the hot region and removing it from the cold region as shown in figure 2. The heat flux J can be calculated by J=
(1)
dq , 2Adt
(2)
where J is the heat flux, A is the cross-sectional area and dt is the time step. The factor of 2 is due to the fact that the heat current propagates in two opposite directions from the hot region to the cold region. The sample is divided into N slabs to evaluate the temperature gradient. The temperature T in each slab is calculated as the average temperature of the all atoms in the slab
where rij is the distance between atoms i and j. Values of σ and ε for the L–J potential are listed in table 1 [35]. In the table, r0 is the distance at which the potential reaches its minimum. We assumed and verified that the interatomic interactions in the L–J potential between atoms beyond two layers are so weak that they contribute little to the total thermal conductivity. This is because their distance is larger than 12 Å which is the cut-off distance for the L–J potential. Thus, we assign a number to each line of atoms as shown in figure 1, which provides an easy way to define the L–J potential in MD simulations for multilayer MoS2. The multilayer MoS2 is stacked in A–A′ sequence, which is the most stable stacking order with the lowest energy [36], where Mo(S) atoms of the second layer are right above S(Mo) atoms of the first layer. The non-equilibrium molecular dynamics (NEMD) simulations are performed to calculate the thermal conductivity. The schematic model setup is shown in figure 2. The equations of atomic motion are integrated with a time step of 1.0 fs. Before applying heat flux, the initial relaxed
N
T=
1 ∑mi vi2 , 3Nk i = 1
(3)
where N is the total number of atoms in the slab, mi is the mass of atom i, vi is the velocity of the atom i and k is the Boltzmann constant. We run 106 time steps to reach steady state and another 4 × 106 time steps to obtain the time-average temperature profile, which is used to calculate the temperature gradient ∂T ∂l, where l is the distance along the heat transfer direction. The thermal conductivity λ can be calculated as λ=
2
J dq = . ∂T ∂l 2Adt (∂T ∂l)
(4)
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Figure 2. Schematic setup of the non-equilibrium molecular dynamics simulation. The width of the hot region is twice of each cold region. At
each time step, a small amount of heat dq is added to the hot region and removed from cold regions to ensure the conservation of energy.
3. Results and discussion 3.1. In-plane thermal conductivity
Here, we focus on the in-plane thermal conductivity in the armchair direction (see figure 2). Periodic boundary conditions are applied in the two in-plane directions. Firstly, we study the thermal conductivity of monolayer MoS2. It is well-known that thermal conductivity predicted by NEMD usually shows strong size effect due to the phonon scattering at sample ends, especially when the sample size is below the mean-free path (MFP) [37, 38]. To obtain the thermal conductivity at infinite sample length, an inverse fitting procedure [31, 37, 39, 40] is employed. We choose the monolayer MoS2 samples with a fixed width of 10 nm and vary the length from 40 to 100 nm. Consequently, the corresponding change of the distance between the hot and cold region is from 20 to 50 nm. As shown in figure 3, the thermal conductivity at infinite length λ∞ can be obtained by linear fitting the computed thermal conductivity at different sample sizes to the following equation [39] 1 1 ⎛ L⎞ ⎜1 + ⎟, = d⎠ λ λ∞ ⎝
Figure 3. The inverse of thermal conductivity for monolayer MoS2
as a function of the inverse of the distance d between the hot and cold region. The thermal conductivity at infinite sample length can be obtained by linear extrapolating to 1/d = 0.
(5)
where λ∞ is the converged thermal conductivity at infinite length, L is the MFP, d is the distance between hot and cold region and λ is the size-dependent thermal conductivity. From the interception of the fitting line at the vertical axis, the λ∞ of the monolayer MoS2 sheet is found to be about 19.76 W mK−1, which is comparable with the previous calculated value of 23.2 W mK−1 by density functional perturbation theory [29] and close to the measured value of 34.5 W mK−1 [41]. We then proceed to study the layer effect on the thermal conductivity. We generated the MoS2 samples with the same in-plane dimensions of 20 × 20 nm2, but with different number of layers. The calculated thermal conductivity as a function of the number of layers is shown in figure 4. It can be observed that the thermal conductivity almost remains constant when the number of layers increases. This is consistent
Figure 4. In-plane thermal conductivity of MoS2 as a function of the number of layers. The thermal conductivity remains almost unchanged as the number of layers increases.
with a recent experiment, which found that the thermal conductivity of four-layer and seven-layer MoS2 are almost the same at various temperatures [30]. However, this effect is quite different from that of few-layer graphene. Experiment has shown that the thermal conductivity of few-layer graphene decreases monotonically with the increase of layer number [22]. This experiment result has also been confirmed by theoretical studies [13, 22, 25]. The reduction in thermal 3
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Figure 6. Cross-plane thermal conductivity of MoS2 as a function of temperature.
case of MoS2, there are some chemical bonds along the outof-plane direction. These out-of-plane bonds have similar chemical properties as the in-plane bonds, making multilayer MoS2 somehow more isotropic in all the directions. As a result, the peaks are similar for both in-plane and out-of-plane DOS because the vibration modes are similar in these two directions as shown in figure 5(b). Figure 5. Vibrational DOS of (a) monolayer graphene and (b)
3.2. Cross-plane thermal conductivity
monolayer MoS2.
We move on to study the cross-plane thermal conductivity of MoS2. In the current study, we focus on the variation of the thermal conductivity and thermal resistance with respect to strain and temperature. The thermal resistance R is defined as:
conductivity in few-layer graphene was attributed to the increased phonon–phonon scattering channels, which are increased by increasing layer number due to the cross-over of the acoustic and optical phonon branches in the phonon spectrum of graphene. In other words, this effect is directly related to the specific phonon spectrum of graphene, which has no energy band gap between the acoustic and optical branches. However it has been shown that the phonon spectrum in MoS2 is characterized with a finite energy gap between the acoustic and optical branches [42]. As a result, the phonon–phonon scattering channels in MoS2 are almost unchanged by increasing layer number, resulting in a layerindependent thermal conductivity. In addition, for graphene, it is assumed that there is no scattering from the top and bottom surfaces since it is only one atomic layer thick, which is too thin to have any cross-plane velocity component [14, 22]. Therefore, the surface roughness parameter p in Ziman formula was set to 1 for the top and bottom surfaces in previous studies [14, 22]. However, due to the intrinsic trilayer structure in each MoS2 layer, the surface roughness of the top and bottom surfaces is important even for monolayer MoS2. As a result, the thermal conductivity in monolayer MoS2 is close to that in multilayer MoS2. This point could be illustrated by the difference in the phonon power spectrum of graphene and MoS2. The vibrational density of states (DOS) are calculated by taking the Fourier transform of the velocity auto-correlation function and the AIREBO potential [43] is adopted for the interatomic interaction in graphene. For graphene, the peak of the total DOS is almost consistent with the peak of in-plain DOS and the shapes of total DOS and inplane DOS are quite close, as shown in figure 5(a). For the
R = D/ λ ,
(6)
where D is the interlayer distance. In the present simulation, the interlayer distance D is chosen as 0.621 nm [36]. 3.2.1. Effect of temperature. Although the in-plane thermal
conductivities of MoS2 at different temperatures have been investigated [34, 44], the effect of temperature on the crossplane thermal conductivity has not been reported. Here we varied the temperature from 100 to 500 K, and the simulated cross-plane thermal conductivity of MoS2 is shown in figure 6. It can be seen that the cross-plane thermal conductivity decreases as the temperature increases from 100 to 500 K. It can also be seen that the temperature-induced reduction is quite significant: the thermal conductivity at 500 K is only about 30% of that at 100 K. 3.2.2. Effect of cross-plane strain. To study the strain effect
on the cross-plane heat transport in MoS2, we choose a model of 30-layer structure with a cross-sectional area 4 × 4 nm2 as shown in figure 7. After the model is fully relaxed, a crossplane strain is applied on the equilibrium structure. The strain is defined as ε = ( l − lo )/ lo,
(7)
where lo and l are the length of the simulation box in the cross-plane direction for the unstrained and strained MoS2 sample, respectively. Note that the strain is applied by 4
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Figure 7. Simulation model of 30-layer MoS2 for studying the cross-plane thermal conductivity. The orientation is indicated by the coordinate. The heat flux is applied in the cross-plane (z) direction.
Figure 8. Mechanical response during the straining process. (a) and (b) stress components induced in all directions. (c) and (d) standard
deviation of the z-coordinate of S and Mo atoms. (e) and (f) distance between S atoms in the same MoS2 layer (SS) and in the two adjacent layers (SS′).
deforming the simulation box size in the cross-plane direction at a constant strain rate of 5 × 108 s−1 while keeping the box size along the other two directions unchanged (NVT ensemble). Figure 8 presents the mechanical response of multilayer MoS2 upon the application of the strain. When compressive strain εc is applied, the stress components, Sxx, Syy, Szz, increase monotonically, while a much bigger stress is induced in the z direction and the stresses in the x and y direction are almost the same (figure 8(a)). It should be noted that the presence of the in-plane stress is due to the trilayer structure of MoS2, while no in-plane stress is generated for graphene since graphene only consists of one atom layer [45]. When tensile strain εc is applied, the stress components increase monotonically till the strain reaches around 0.07, at which the
tensile strain overcomes the L–J potential, the system breaks and the stress components return near zero, as shown in figure 8(d). Moreover the standard deviations of the zcoordinate of S atom (S_S) and Mo atom (S_Mo) remain at very low values during compressing or stretching process, as shown in figures 8(b) and (e), which indicate that no buckling occurs. In addition, the variations of the distance between S atoms in the same layer (SS) as well as in two adjacent layers (SS′) (see figure 9) are also calculated and shown in figures 8(c) and (f). It can be seen that the SS remains almost unchanged as compared with SS′, especially during the stretching process. This is because the bonding interaction between atoms in the same MoS2 layer (SW potential) is much stronger than the non-bonding van der Waals interaction (L–J potential). It is noted that there is some difference in 5
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Figure 9. Schematic illustration to the definition of distance between S atoms in the same MoS2 layer SS and in the two adjacent layers SS′.
SS′ between figures 8(c) and (f). This difference in SS′ during the compressing and stretching process is caused by the asymmetry of the L–J potential. The cross-plane thermal conductivity and resistance at room temperature under various cross-plane strains are shown in figure 10. The calculated values are normalized by the corresponding values at zero strain. It is seen that with a 10% compressive strain, which corresponds to a compressive stress of about 14 GPa, is able to enhance the thermal conductivity more than 14 times and reduce the thermal resistance by 94%. A 5% tensile strain, which induces about 1.5 GPa tensile stress, is able to reduce the thermal conductivity by nearly 90% and increase the thermal resistance by a factor of 10. It is noted that this trend is in good agreement with the simulation result of bulk argon using L–J potential [46, 47]. To understand this trend, the vibrational DOS are calculated as shown in figure 11. From the kinetic theory, the thermal conductivity λ can be expressed as λ=
∑Ci vi L i,
Figure 11. Vibrational DOS of multilayer MoS2 under different cross-plane strains. (a) Out-of-plane vibrational DOS. (b) In-plane vibrational DOS. (c) Total vibrational DOS.
The enhanced force constant increases the phonon frequency, thus causing the blue shift of the major peaks in the phonon spectrum as shown in figure 11. Such a blue shift results in an increase in both group velocity and specific heat [48, 49], as a result, the thermal conduction increases according to equation (8). Similarly, tensile strain softens the phonon
(8)
i
where Ci, vi, Li are, respectively, the specific heat, group velocity and MFP of phonon mode i. When compressive strain is applied, the interlayer L–J interaction gets enhanced.
Figure 10. (a) Cross-plane thermal conductivity and (b) thermal resistance as a function of cross-plane strain. 6
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Figure 12. Mechanical response during the straining process. (a) and (b) stress components induced in all direction. (c) and (d) standard deviation of the z-coordinate of S atom and Mo atoms. (e) and (f) distance between S atoms in the same MoS2 layer SS and in two adjacent layers SS′.
Figure 13. Variations of (a) cross-plane thermal conductivity and (b) cross-plane thermal resistance as a function of biaxial in-plane strain.
standard deviations of the z-coordinate of S atoms (S_S) and Mo atoms (S_Mo). It was shown that the thermal transport behavior of buckled structure is quite different from that of flat ones [49], hence, it is important to check whether buckling occurs within the strain range. We find that the values of S_S and S_Mo in figures 12(b) and (e) remain at very low values during compressing, thus buckling does not occur within the compression strain range. We also double checked the atomic configuration of the compressed structure at 10% compressive strain and no buckling was observed. The variations of the cross-plane thermal conductivity and resistance as a function of the in-plane strain are shown in figure 13. The tensile strain results in an increase of thermal resistance and a decrease of thermal conductivity, while the compressive strain leads to a decrease of thermal resistance and an increase of thermal conductivity. This effect of inplane strain on the cross-plane thermal conductivity and
vibration and results in a red shift of the major peaks, which reduces the thermal conductivity. In addition, the inter-layer distance d decreases during the compressing process and increases during the stretching process, thus resistance R decreases when compressive strain is applied and increases when tensile strain is applied as shown in figure 10(b). 3.2.3. Effect of in-plane strain. We then move on to study the
effect of in-plane strain on the cross-plane thermal transport. Biaxial strain is applied in the x and y directions of the simulation model (the in-plane of MoS2), while the model size in the z direction is fixed. When biaxial compressive strain is applied, compressive stresses in all directions increase significantly, see figure 12(a). When tensile strain is applied, tensile stresses in the x and y directions increase significantly while the stress in the z direction shows a much smaller increase, see figure 12(d). We also calculated the 7
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4. Conclusion We studied the in-plane and cross-plane thermal conductivities of MoS2 by using NEMD. A value of around 19.76 W mK−1 for the in-plane thermal conductivity is obtained. We further found that the in-plane thermal conductivity of multilayer MoS2 is insensitive to the number of layers, which may be explained by the finite energy gap between the acoustic and optical branches in the phonon spectrum and its intrinsic trilayer structure. The cross-plane thermal conductivity is found to decrease when the temperature rises from 100 to 500 K. Moreover, we also studied strain effect on the cross-plane thermal conductivity and found that the cross-plane strain has a much stronger effect than the in-plane strain. A 10% cross-plane compressive strain can enhance the thermal conductivity by a factor of 10, while a 5% cross-plane tensile strain can reduce the thermal conductivity by 90%. Our findings may be useful for thermal management in MoS2 based nanodevices and for thermoelectric applications of MoS2.
Acknowledgments The work is supported by the Agency for Science, Technology and Research (A*STAR), Singapore and by the Recruitment Program of Global Youth Experts of China and start-up funding from the Shanghai University.
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