Manipulating the Thermal Conductivity of Monolayer MoS2 via Lattice

Article pubs.acs.org/JPCC

Manipulating the Thermal Conductivity of Monolayer MoS2 via Lattice Defect and Strain Engineering Zhiwei Ding,† Qing-Xiang Pei,*,† Jin-Wu Jiang,*,‡ and Yong-Wei Zhang*,† †

Institute of High Performance Computing, A*STAR, 1 Fusionopolis Way, Singapore 138632, Republic of Singapore Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, China

‡

ABSTRACT: Monolayer molybdenum disulfide (MoS2), a new two-dimensional material beyond graphene, has attracted tremendous attention recently. Its applications in nanoelectronic and thermoelectric devices usually require manipulating the thermal transport properties. Using nonequilibrium molecular dynamics simulations, we investigated the effects of lattice defects and mechanical strain on the thermal conductivity of MoS2. We found that the thermal conductivity of monolayer MoS2 can be effectively tuned by introducing even a small amount of lattice defects. For example, a 0.5% concentration of mono-Mo vacancies is able to reduce the thermal conductivity by about 60%. Remarkably, the thermal conductivity of the defected sample can further be tuned by mechanical strain. For example, a 12% tensile strain is able to reduce the thermal conductivity by another 60%. We also found that the tensile strain exerts almost the same impact on the thermal conductivity of both pristine and defective MoS2, which signifies that there is no apparent coupling between defects and strain in affecting the thermal conductivity. Our analyses of the vibrational density of state and spectral energy density show that the underlying mechanisms for these drastic changes are (1) the reduction of the phonon relaxation time arising from phonon−defect scattering and (2) the reduction of the group velocity and heat capacity caused by tensile strain. Our findings here provide important insights and guidelines for the use of monolayer MoS2 in thermal management and thermoelectric devices.

1. INTRODUCTION Two-dimensional monolayer molybdenum disulfide (MoS2) has attracted great attention recently due to its unique electronic,1 optical,2 and mechanical3−5 properties. Unlike graphene, which is a gapless conductor, monolayer MoS2 is a semiconductor with a sizable direct band gap,6 which makes it a promising candidate for many nanoelectronic and optoelectronic applications.7−10 In addition, MoS2 has great potential to be used in thermoelectric devices due to its high Seebeck coefficient.11 It is well-known that the performance of these devices strongly relies on material quality. Currently, it still remains a challenge to synthesize large-size high-quality monolayer MoS2.12 Structure defects including vacancies, dislocations, grain boundaries, and edges in MoS2 can either be formed during growth process13−15 or introduced by electron irradiation.16,17 It has been both theoretically predicted13,18 and experimentally observed12 that those defects are able to significantly affect the electronic transport by generating defect-induced localized states.13 Consequently, defect engineering on MoS2 presents promising opportunities to tune their properties or even create new functionalities. For example, point defects in MoS2 can trap free charge carriers and localize excitations, leading to a new photoemission peak and enhancement in photoluminescence intensity.19,20 Strain-induced magnetism in defective monolayer MoS2 assures an opportunity in the design of magnetic switching or logic devices.21 Defect-induced aniso© XXXX American Chemical Society

tropy in thermal and electrical conductivities enables defect engineering as a powerful tool in current management.22 Given the fact that defects are unavoidable in MoS214 and defected monolayer MoS2 has to be used in electronic devices,19,20 it is also necessary to study the effects of defects on the thermal conductivity of MoS2, which is important for the heat dissipation in MoS2-based nanodevices and for the thermoelectric applications of MoS2.11 It is noted that previous studies on the thermal conductivity of MoS2 were limited to pristine MoS2 structures,23−29 though intensive studies can be found on the effect of defects on the thermal conductivity of other materials, such as graphene,30−38 silicene,39−41 silicon nanowires,42,43 carbon nanotubes,44,45 hexagonal boron nitride,46 and diamond.47 Up to now, there is no report on the effect of defects on the thermal conductivity of MoS2. Defects can also serve as a means to modify the thermal conductivity of materials due to defect-induced phonon scattering.31,39,48 Another means of controlling thermal transport is through applying mechanical strain.49−53 For example, the thermal conductivity was shown to decrease continuously when the strain changed from compressive to tensile for bulk silicon and diamond, silicon and diamond nanowires, and silicon and diamond thin films.50 However, for carbon Received: April 15, 2015 Revised: June 3, 2015

A

DOI: 10.1021/acs.jpcc.5b03607 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

The heat conduction region was divided into N slabs in the x direction to calculate the temperature in each slab and evaluate the temperature gradient. The temperature T in each slab was calculated as the average temperature of all atoms in the slab

nanotubes and graphene, the thermal conductivity showed a reduction at large compressive strains due to structure buckling-induced stronger phonon−phonon scattering.51,52 A very recent study showed that tensile strain could obviously reduce the thermal conductivity of pristine MoS2.27 However, it is still unclear how the strain will affect the thermal conductivity of defective MoS2. In this work, we demonstrate by using molecular dynamics (MD) simulations that the thermal conductivity of MoS2 could be drastically reduced with a very low concentration of defects. More specifically, 0.5% Mo (S) vacancy defects could reduce the thermal conductivity by about 60% (35%). Moreover, we show that the thermal conductivity of the defected MoS2 can be further drastically reduced by mechanical strain, and 12% tensile strain can reduce the thermal conduction of the defected MoS2 by more than 60%. The underlying mechanisms for the significant reduction in thermal conductivity are explored based on the analysis of the vibrational density of state (DOS) and spectral energy density (SED).

T=

1 3NkB

N

∑ mi v i2

(2)

i=1

where N is the total number of atoms in the slab, mi is the mass of atom i, and kB is the Boltzmann constant. We ran 106 time steps to reach steady state and another 2 × 106 time steps to obtain the time-average temperature profile, which was used to calculate the temperature gradient ∂T/∂x. The thermal conductivity K in the x direction can be calculated as K=−

Jx ∂T /∂x

(3)

where Jx is the heat flux in the x direction. 2.2. Analysis Methods. 2.2.1. Vibrational DOS. The vibrational DOS,g(ω), can be calculated from the Fourier transformation of the atomic velocity autocorrelation functions (VACFs) G(t), which is expressed as

2. SIMULATION MODEL AND METHODS 2.1. Simulation Models. All MD simulations in our study are performed using the LAMMPS package.54 The Stillinger− Weber (SW) potential27 is used to describe the interatomic interactions in MoS2. The schematic of the simulation model is shown in Figure 1. The periodic boundary condition was used

G(t ) = ⟨vi(τ ) ·vi(τ + t )⟩i , τ =

1 N

N

∑ i=1

1 t int − t

∫0

t int − t

vi(τ ) ·vi(τ + t ) dτ

(4)

where tint is the time of integration, N is the total number of atoms in the ensemble, vi is the velocity vector of the ith atom, and ⟨ ⟩ stands for the ensemble average. Thus, g(ω) can be evaluated as

Figure 1. Schematic of the model for nonequilibrium MD simulation.

g (ω) =

∫0

=

∫0

t int

t int

G(t )e−iωt dt ⟨vi(τ ) ·vi(τ + t )⟩i , τ e−iωt dt

(5)

In calculating the DOS, tint should be an order of magnitude longer than the longest phonon relaxation time, and the sampling rate should be greater than twice the maximum allowed frequency. In our MD simulations, atomic velocities were recorded every 25 fs for 51.2 ns. 2.2.2. Phonon SED. For convenience, we use the orthogonal unit cell in our MD simulations to calculate the SED function, as indicated in Figure 2. There are 6 atoms in the unit cell; therefore, there should be 18 branches of dispersion curves. The SED analysis can predict phonon dispersion relations and lifetimes directly from velocity of atoms in a crystal using the phonon SED function.57 The SED is evaluated by converting spatial and time-dependent atomic velocity data into frequency and reciprocal space. The SED function Φ(k,ω) is expressed as

in the x (longitudinal) and y (transverse) direction, and the free boundary condition was applied in the z (out-of-plane) direction. The nonequilibrium molecular dynamics (NEMD) simulations were performed to calculate the thermal conductivity. The equations of atomic motion were integrated with a time step of 0.5 fs. The dimension of the monolayer MoS2 sample was 12 × 12 nm2. The initial relaxed configuration was equilibrated at 300 K under constant pressure and constant temperature (NPT) for 105 time steps. After NPT relaxation, the regions at the two ends of the simulation model along the heat conduct direction (see Figure 1) were fixed in location. To establish the temperature gradient, the hot and cold regions were introduced by applying a Langevin thermostat55 of temperatures 350 and 250 K, respectively. The part between the hot and cold regions is the heat-conducting region. The constant volume and constant energy (NVE) ensemble was applied to the heat-conduction region. The heat flux vector was calculated based on contributions from atoms in the heat conduction region. The heat flux J was calculated by 1 J = [∑ Ei vi − ∑ Sivi] V i (1) i

Φ(k, ω) =

1 4πt intNT

B

t int NT

∑ ∑ mb|∫ ∑ vα(ni , b , t ) α

b

0

ni

× exp[ik·r(ni) − iωt ] dt |2

(6)

where NT is the total number of unit cells, B is the number of atoms per unit cell (B = 6 for the unit cell chosen), and vα(ni,b,t) is the velocity in the α direction of atom b (with mass mb) inside of the unit cell ni. In this study, we focus on the thermal conductivity in the armchair direction; thus, the simulation model consists of 100 unit cells in the x (armchair)

where Ei and vi are the total energy (both kinetic and potential) and the velocity of atom i and Si is the stress tensor.54,56 B


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Figure 4. Variation of thermal conductivity of monolayer MoS2 with respect to the concentration of defects. Red rectangle: MSV; green circle: MMoV; blue triangle: DSV; black triangle: SI. The dashed lines are the fitting curves.

Figure 2. Schematic of the (a) top view and (b) side view of monolayer MoS2. The rectangle shows the selection of the unit cell for the SED analysis. As there are 6 atoms (4 S atoms and 2 Mo atoms) in the unit cell, there will be 18 branches of dispersion curves, of which the first 6 are acoustic.

4, the thermal conductivity of MoS2 decreases rapidly with increasing defect concentration for the different types of defects, in particular, for the MMoV defect. At the defect concentration of 0.5%, the thermal conductivity is reduced by 35% with MSV or DSV defects, while it is reduced by about 60% with MMoV or SI defects. This significant reduction in thermal conductivity of MoS2 presents a great opportunity for thermoelectric applications. Fitting the simulated thermal conductivities in Figure 4 gives KMSV 1 = K0 1 + 1.1358f (8)

direction and 5 unit cells in the y (zigzag) direction, and we focus on the SED function of wave numbers in the x direction. The relaxation time of the phonon with a wavenumber kx and frequency ω can be obtained by fitting the SED to a Lorentzian function58 I Φ(k x, ω) = 1 + [2τk x, ωc(ω − ωc)]2 (7) where I is the peak magnitude, ωc is the frequency at the peak center, and τkx,ωc is the relaxation time of mode (kx,ωc).

KMMoV 1 = K0 1 + 4.3799f

3. RESULTS AND DISCUSSION First, we study the thermal conductivity of defective MoS2. We focus on four types of defects, (1) mono-S vacancy (MSV), (2) double-S vacancy (DSV), (3) mono-Mo vacancy (MMoV), and (4) S interstitial (SI), as shown in Figure 3. The defects are randomly distributed in the structure. These defects are generated by removing atoms from or adding atoms to the well-relaxed pristine MoS2 sheet. We define the defect concentration f as the ratio of the number of atoms removed/added over the total number of atoms in the pristine MoS2 sheet. The effects of each defect type on the thermal conductivity are investigated separately by performing NEMD simulations for various defect concentrations. As seen in Figure

(9)

KDSV 1 = K0 1 + 1.0982f

(10)

KSI 1 = K0 1 + 3.1005f

(11)

where K0 is the thermal conductivity of pristine MoS2 at 300 K and f is the defect concentration. Similar expressions were obtained for defective graphene32 and silicon,59 and the parameter before f was 5.718 and 1.376, respectively. Thus, we conclude that the sensitivity of thermal conductivity of the MoS2 to S vacancy is comparable to that of silicon system,

Figure 3. Schematic of three types of defects: (a) MSV, (b) DSV, (c) MMoV, and (d) SI. The atoms inside of circles are undercoordinated atoms near vacancies. C


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Figure 5. Comparison of the vibrational DOSs between (a) pristine and 0.25% MSV MoS2, (b) pristine and 0.25% MMoV MoS2, (c) pristine and 0.25% DSV MoS2, and (d) pristine and 0.25% SI MoS2.

Figure 6. Phonon dispersion relationship of (a) pristine MoS2, (b) defective MoS2 with 0.5% MSV, and (c) defective MoS2 with 0.5% MMoV. Dispersion curves almost remain the same when defects are induced, but a clear broadening effect can be observed especially for Mo defects, which indicates the reduction in the MFP and relaxation time.

Figure 7. Relaxation time of selected phonons for pristine and defective MoS2.

conductivity becomes more and more insensitive to defects with increasing defect concentration. From the kinetic theory, the thermal conductivity K can be expressed as

while the sensitivity to the Mo vacancy is similar to that of the graphene system. On the basis of eqs 8−11, we see that the thermal conductivity of a pristine monolayer MoS2 can be reduced to half of its value by introducing MMoV defects at the concentration of 0.23% or MSV (DSV) defects at the concentration of 0.9% or SI defects at the concentration of 0.32%. It can also be seen from Figure 4 that the thermal conductivity is more sensitive to defects at smaller defect concentrations. For example, when a 0.12% concentration of MSVs is introduced into pristine MoS 2 , the thermal conductivity reduces by 38%; however, when the defect concentration increases from 0.12 to 0.5%, that is, a further addition of 0.38% concentration of MSVs, there is only 22% reduction in thermal conductivity. Clearly, the thermal

K=

∑ CivL i i i

(12)

where Ci, vi, and Li are, respectively, the specific heat, group velocity, and mean free path (MFP) of phonon mode i. In order to find out the effect of defects on the phonon properties, the vibrational DOSs of pristine and defective MoS2 at 0.25% defect concentration were calculated and are shown in Figure 5. It is seen that the difference in the shape and peaks of the DOSs between pristine and defective MoS2 is very small, indicating that the defects cause a minor change in the phonon specific heat and group velocity.32 Thus, we can deduce from eq D


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between the undercoordinated atoms near the vacancies.60 Adopting the Mathiessen’s rule, which assumes the independence of different scattering mechanisms, the total scattering rate τ−1 is given as τ −1 = τU−1 + τB−1 + τV−1 + τA−1

(13)

τ−1 U

where is the Umklapp phonon−phonon scattering rate, τ−1 B is the phonon−boundary scattering rate, τ−1 V is the scattering rate due to the missing mass and linkages, and τ−1 A is the scattering rate due to the change in the force constant of bonds. 61,62 The scattering rate τ−1 V can be expressed as ⎛ ΔM ⎞2 π ω 2g (ω) ⎟ τV−1 = f ⎜ ⎝ M ⎠ 2 G

(14)

where f is the defect concentration, G is the number of atoms, and g(ω) is the phonon DOS. The effective value of ΔM/M is −(Ma/M) − 2, where M is the average mass per atom, Ma is the mass of the missing atom, and the term −2 accounts for the potential energy of the missing linkages.60,62 As the atomic masses of the Mo and S atoms are 96 and 32, respectively, the scattering rate resulting from the missing mass and linkages (τ−1 V ) by Mo vacancies is greater than that by S vacancies according to eq 14. This is consistent with the observation in Figure 4 that thermal conductivity is more sensitive to Mo vacancies than S vacancies as the changes in g(ω) and G caused by vacancies can be ignored due to the very low concentration of vacancies in our study. 63 The scattering rate τ−1 A can be expressed as Figure 8. Variation of the thermal conductivity of defective MoS2 as a function of mechanical strain. The thermal conductivity is normalized by the thermal conductivity of (a) strain-free pristine MoS2 and (b) strain-free defective MoS2. The defect concentration of the samples is 0.25%.

⎛ δk ⎞2 ω 2g (ω) τA−1 = nf ⎜ ⎟ 4π ⎝k ⎠ G

(15)

where n is the number of undercoordinated atoms near a vacancy, k is the force constant, and δk is the change in the force constant. As shown in Figure 3, n is 3 and 6 for MSV and MMoV, respectively. Due to the greater atom number and atom mass, δk caused by the Mo vacancy is greater than that from the S vacancy. Thus, the scattering rate due to the change in force constant of bonds between the undercoordinated atoms near the vacancies (τ−1 A ) introduced by the Mo vacancy is greater than that by the S vacancy according to eq 15, which is also consistent with the observation that the thermal conductivity is more sensitive to the Mo vacancy. Next, we study the effect of mechanical strain on the thermal conductivity of defective monolayer MoS2. Previous studies showed that mechanical strain could reduce the thermal conductivity of pristine MoS2 and other materials, such as silicon and diamond (bulk, thin films, and nanowires),50 carbon nanotubes,50,51 and graphene.50,52,64,65 The reduction in thermal conductivity was 20−30% for silicon nanowires and thin films when a 12% tensile strain was applied.50 A 8% tensile strain reduced the thermal conductivity of pristine MoS2 by 35%.27 However, it was also reported that tensile strain resulted in an increase in the thermal conductivity of amorphous silicon thin films,66 polymer chains,67 and silicene.49,53 Therefore, the effect of mechanical strain on the thermal conductivity of MoS2 needs to be studied in detail, in particular, in the presence of defects, as the work on the combined effect of stain and defects on the thermal conductivity of materials is still lacking. Here, we apply uniaxial strain along the heat flux direction and calculate the thermal conductivity in that direction. Because fracture will occur in the defective structure when the tensile strain is larger than 12%, and severe structure buckling will

12 that the reduction in thermal conductivity due to the defects is mainly attributed to the reduction in the phonon MFP. From the enlarged selected peaks, it is seen that slight broadening of the mode peaks is visible. This broadening indicates a reduction in relaxation time of the corresponding mode, which reduces its phonon MFP and contribution to the thermal conductivity. To further illustrate this point, we plot the natural logarithm of the SED for both pristine and defective MoS2 in Figure 6. It can be seen that the shape of the dispersion curves remains almost unchanged when defects are introduced in monolayer MoS2, which confirms the minor change in the specific heat and group velocity. However, compared with pristine MoS2, a broadening in the dispersion curves of defective MoS2 is apparent, especially for Mo vacancies. This broadening of the dispersion curves indicates the reduction in the MFP and relaxation time in defective MoS2. The relaxation time can be obtained using eq 7. The relaxation times of phonons with wavenumbers kx = 0.2 and 0.25 and frequency less than 6 THz are plotted in Figure 7. We choose these phonons because (1) the dispersion curves at these wave vectors are distinguishable and (2) low-frequency phonons are the major heat carriers. It can be seen from Figure 7 that although there are some exceptional cases, the relaxation time of phonons is generally reduced when defects are introduced. Besides, the relaxation time is more sensitive to Mo vacancy defects. The impact of vacancies on phonon relaxation time is due to phonon scattering by vacancies arising from missing mass, missing linkage, and the difference in force constants of bonds E


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Figure 9. Vibrational DOS of MoS2 under different tensile strains for (a) pristine, (b) 0.25% Mo vacancy, (c) 0.25% S vacancy, and (d) 0.25% SI.

Figure 8b shows that the tensile strain exerts almost the same impact on the thermal conductivity of both pristine and defective MoS2, which signifies that there is no apparent coupling between defects and strain in affecting the thermal conductivity. This can be understood from the different mechanisms in which strain and defects affect the thermal conductivity. Defects play the critical role as scattering centers in reducing the phonon relaxing time, as indicated in Figures 6 and 7, but they have little effect on the group velocity and heat capacity because only a minor change in the DOS is observed when defects are introduced, as elaborated on in Figure 5. In contrast, strain mostly affects the heat capacity and group velocity by softening/stiffening the phonons. When a compressive strain is applied on MoS2, the thermal conductivity decreases with increasing compressive strain (see Figure 8). A 4% compressive strain can lead to a reduction in the thermal conductivity by ∼10%. This behavior is different from that of traditional bulk materials,68 whose thermal conductivity increases with increasing compressive strain due to stiffening of phonon modes. The compressive strain-induced reduction of the thermal conductivity in MoS2 can be attributed to the out-of-plane deformation of the monolayer structure, as shown in Figure 10b, which enhances the phonon scattering and thus reduces the thermal conductivity. A similar reduction effect was also observed in kinked silicon nanowires,69 compressed carbon nanotubes,51 and graphene.52

Figure 10. Snapshots of the MoS2 sheet at (a) strain-free and (b) 2% compressive strain.

occur when the compressive strain is beyond −4%, the strain range in the present study is chosen from −4 to 12%. We study three defective MoS2 sheets: (1) 0.25% MSV defects, (2) 0.25% MMoV defects, and (3) 0.25% SI defects. The calculated thermal conductivities of the defective MoS2 at different mechanical strains are shown in Figure 8. It can be seen that the tensile strain exerts a strong influence on the thermal conductivity of defective MoS2. With the increase of tensile strain, the thermal conductivity decreases monotonically. At the tensile strain of 12%, the thermal conductivity of defective MoS2 drops by more than 60%. Hence, the tensile strain provides an effective means to tune the thermal conductivity of defective monolayer MoS2. The variation of thermal conductivity with tensile strain can be understood from the vibrational DOSs. It is seen from Figure 9 that the tensile strain softens the phonon vibrations and results in a red shift of major peaks in the vibrational DOSs. Such a red shift can result in a decrease in both group velocity and specific heat,50,51 which leads to the reduction in thermal conductivity according to eq 12.

4. CONCLUSIONS In summary, we have studied the effects of lattice defects and tensile strain on the thermal conductivity of defective MoS2. It is found that the thermal conductivity can be effectively tuned by changing defect concentrations. Specifically, a 0.5% F


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concentration of Mo (S) defects can reduce the thermal conductivity by about 60% (35%). Through vibrational DOS and SED analysis, we reveal that the phonon relaxation time is the crucial factor accounting for the reduction of the thermal transport in defective MoS2. Furthermore, we find that the thermal conductivity of defective MoS2 can be further tuned by applying tensile strain. A 12% tensile strain can reduce the thermal conductivity by more than 60% and the strain. We also found that tensile strain exerts almost the same impact on the thermal conductivity of both pristine and defective MoS2, which signifies that there is no apparent coupling between defects and strain in affecting the thermal conductivity. Our findings are important for thermal management in MoS2-based nanodevices and thermoelectric applications of MoS2.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Telephone: +65 64191225. (Q.-X.P.). *E-mail: [email protected]. Telephone: +86 2066001004 (J.-W.J.). *E-mail: [email protected]. Telephone: +65 64191478 (Y.-W.Z.). Notes

The authors declare no competing financial interest.

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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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