Beyond Perturbation: Role of Vacancy-Induced Localized Phonon

Article pubs.acs.org/JPCC

Beyond Perturbation: Role of Vacancy-Induced Localized Phonon States in Thermal Transport of Monolayer MoS2 Bo Peng,† Zeyu Ning,† Hao Zhang,*,† Hezhu Shao,‡ Yuanfeng Xu,† Gang Ni,† and Heyuan Zhu*,† †

Shanghai Ultra-precision Optical Manufacturing Engineering Research Center and Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China ‡ Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China S Supporting Information *

ABSTRACT: Sulfur vacancies in monolayer MoS2 can provide unexpected opportunities for tailoring the properties and device applications via defect engineering. However, determining the effect of vacancies in thermal transport remains a big challenge. Using a first-principles supercell approach, we reveal the dominant role of defect-induced quasilocalized phonon states in reducing thermal conductivity of MoS2. These states are related to flattened dispersions in phonon spectrum, which comes from perturbations in atomic mass and interatomic bonding. Although the scattering strength of each modes remains similar, the phonon group velocities are much lower near the quasi-localized modes, while the Umklapp scattering are significantly enhanced. Thus, the thermal conductivity of defective MoS2 is severely reduced. Our results contribute to fundamental understanding of the effect of vacancies on thermal transport, and can be used to assess the defect concentrations in semiconductors quantitatively.

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monolayer MoS2 can be doped or functionalized by filling the vacancies, which further widen the application in electronics, optoelectronics, and spintronics.7,23,42−44 However, the role of vacancy-induced quasi-localized phonon states in thermal transport of monolayer MoS2, which is closely related to these potential applications, remains unexplored. Experimentally measured thermal conductivities κ are 34.5 ± 4,45 62.2,46 and 84 ± 17 W/mK47 for monolayer MoS2, 5248 and 44−52 W/mK49 for few-layer MoS2, and 110 ± 20 W/mK for bulk MoS2,50 respectively. The measured κ of monolayer MoS2 differs significantly from each other, and is much lower than previous theoretical predictions 83−155 W/mK.51−55 Considering the high vacancy concentration in natural MoS2 (0.1−10%56,57) and the imperfection of different growth processes,15−22 it is important to investigate accurately the effect of sulfur vacancies on the κ of MoS2. Although the standard molecular dynamics simulations provide insights into the mass and interatomic force constant perturbations by point defects,58,59 the effect of defects on anharmonic scattering of phonons is not known. Other lifetime models such as Klemens expression that adjusted to reproduce the experimental data agree well with measurements,60,61 but their predictive ability is limited. Recently, fully microscopic computational materials techniques have been developed to study the phonon transport

INTRODUCTION Monolayer MoS2 is currently being object of great attention due to its unique properties. Compared to an indirect bandgap of 1.3 eV in the bulk form where the adjacent layers are connected by the van der Waals interaction,1,2 monolayer MoS2 has a direct semiconducting gap of 1.9 eV, which is promising for field-effect transistor with large on/off ratio as well as for optoelectronic applications.3−7 In addition, the combination of spin−orbit coupling and optical activity makes monolayer MoS2 promising for spintronic applications.8−14 However, due to the imperfection of the growth processes,15−19 the point defects in monolayer MoS2 are particularly noticeable, especially sulfur vacancies.20−22 The defect formation mechanisms including the formation pathway for different sulfur vacancies have been investigated in detail.23−25 Point defects such as vacancies usually play a decisive role in the physical properties of materials.26 For instance, sulfur vacancies induce localized donor electronic states in the bandgap, leading to hopping transport under low carrier densities.3,20−22,27−29 In fact, although high-performance electronic devices require the growth of ultrahigh-quality MoS2, sulfur vacancies can provide unexpected opportunities for tailoring the properties and device applications via defect engineering.30 For example, strong photoluminescence enhancement and wide-spectrum response in defect-engineered MoS2 offer new opportunities to improve the performance of optoelectronic devices,31−35 the sulfur vacancies contribute to significant enhancement of the electrocatalytic performance,36−38 and the magnetic properties of MoS2 can also be tuned by sulfur vacancies.39−41 In addition, © 2016 American Chemical Society

Received: October 27, 2016 Revised: December 7, 2016 Published: December 8, 2016 29324

DOI: 10.1021/acs.jpcc.6b10812 J. Phys. Chem. C 2016, 120, 29324−29331

Article

The Journal of Physical Chemistry C

Figure 1. Schematic of the top and side view of monolayer (a) perfect MoS2 and three types of defects: (b) monosulfur vacancy, (c) disulfur vacancy, and (d) double monosulfur vacancy.

properties,62−65 which offer a useful tool to study the effect of disorder.66−68 In this paper, we calculate the lattice thermal conductivity of perfect and defective MoS2 from first principles. The crystal structures of defective MoS2 are generated by removing atoms from a supercell model. The role of defect-induced quasilocalized phonon states in diffusion and phonon scattering mechanism are all accounted for in the solution of the Boltzmann transport equation for phonons, which reveals the underlying mechanism for the reduction in thermal conductivity of MoS2.

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and can be calculated as a sum of contribution of all the phonon modes λ,51,76 which comprises both a phonon branch index j and a wave vector q κ = καα =

1 V

∑ Cλvλα 2τλα (1)

λ

where V is the crystal volume, Cλ is the heat capacity per mode, and vλα and τλα are the group velocity and relaxation time of mode λ along α direction, respectively. We use the nominal layer thicknesses h = 6.033 Å corresponding to the van der Waals radii of MoS2.51 Using the ShengBTE code,62−65 the κ can be calculated iteratively. The convergence of thermal conductivity with respect to q-mesh is tested in our calculation. The Brillouin zone integrations are carried out on a Γ-centered regular grid of 90 × 90 × 1 and 30 × 30 × 1 q-mesh for perfect and defective MoS2, respectively. Energy-conserving delta functions in the three-phonon processes are approximated by a Gaussian function with a scale parameter of 1 for broadening.64

METHODOLOGY

We present a fully first-principles calculation to investigate the thermal transport properties from the density-functional theory using the Vienna ab initio simulation package (VASP).69,70 The exchange and correlation interactions are incorporated as the generalized gradient approximation (GGA) in the Perdew− Burke−Ernzerhof (PBE) parametrization.71 A plane wave basis with a cutoff energy of 600 eV is employed to represent the electronic wave functions. During the structural relaxation for perfect MoS2, a 21 × 21 × 1 q-mesh is used, until the energy differences are converged within 10−8 eV, with a Hellman− Feynman force convergence threshold of 10−6 eV/Å. A vacuum spacing larger than 15 Å is used to eliminate interactions between adjacent layers. The harmonic interatomic force constants are obtained by calculating the dynamical matrix through the linear response of electron density based on the density functional perturbation theory (DFPT) using the supercell approach.72 For perfect MoS2, a 5 × 5 × 1 supercell with 5 × 5 × 1 q-mesh is used. For defective MoS2, the crystal structures are generated by removing one or two atoms from a 3 × 3 × 1 supercell, and the vacancy concentration (3.7% and 7.4% respectively) is in the range of natural concentration 0.1−10%.56,57 The phonon dispersion and thermodynamic properties are calculated from the harmonic interatomic force constants using the PHONOPY code.73,74 The anharmonic interatomic force constants are calculated using a finite-difference method65 using the same supercell with same q-mesh for perfect and defective MoS2 respectively. The interactions with fifth nearest-neighbor atoms are included for both structures, which is well converged. The electronic contributions to thermal conductivity of semiconducting transition metal dichalcogenides are usually negligible,54,61 and no metallic behavior is observed in defective MoS2 with sulfur vacancies since the presence of sulfur vacancies only introduces localized defect states in the bandgap.21,75 Thus, we only consider phonon transport hereafter. The in-plane lattice thermal conductivity is isotropic

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RESULTS AND DISCUSSION The crystal structures of perfect and defective MoS2 are shown in Figure 1. We focus on three types of defects: (i) monosulfur vacancy (VS), (ii) disulfur vacancy (VS2), and (iii) double monosulfur vacancy (2VS), which are generated by removing atoms from perfect MoS2 supercell. In all samples, VS is most frequently observed,20,22 while VS2 and 2VS have twice the density of sulfur vacancies in VS. The only difference between VS2 and 2VS is the distribution of sulfur vacancies: a S2 column is missing in VS2 (Figure 1c), while for 2VS, two S atoms are missing from two sites of the first layer (Figure 1d). We first investigate the relative stability of defective MoS2 by calculating the defect formation energy Eform = Edefective − Eperfect + nES

(2)

where Edefective and Eperfect are total energy of the 3 × 3 × 1 supercell of defective and perfect MoS2, respectively, n is the number of removed S atoms, and ES is the total energy of S atoms. As listed in Table 1, the defect formation energy for VS is in good agreement with previous result.23 In fact, in highresolution transmission electron microscopy experiments, sulfur vacancies are formed under exposure to an electron beam, which is much easier than molybdenum vacancy formation.23 The lattice thermal conductivity κ of perfect and defective MoS2 as a function of temperature is shown in Figure 2a. The calculated κ at 300 K is listed in Table 1. We also extract the contributions of different phonon branches to κ for phonon engineering. Here the different modes are simply distinguished 29325


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The Journal of Physical Chemistry C Table 1. Calculated Defect Formation Energy Eform of Defective MoS2, Lattice Thermal Conductivity κ at 300 K, and Contribution of Different Phonon Branches (ZA, TA, LA, and All Optical Phonons) toward κ in Perfect and Defective MoS2

perfect VS VS2 2VS

Eform (eV)

κ (W/mK)

ZA (%)

TA (%)

LA (%)

optical (%)

− 6.69 13.84 13.38

154.3 62.1 38.2 40.2

29.0 11.5 9.7 17.2

30.5 23.1 27.1 22.6

39.1 47.6 51.3 40.9

1.4 17.8 11.9 19.3

largest harmonic interatomic force constants in Table 2), leading to lifting the degenerate dispersion branches, as well as shifted and flattened dispersion characteristics, especially at low frequencies. As shown in Table 2, the heat capacity is insensitive to difference in phonon dispersions, because only a small portion of the branches emerge or shift slightly. Therefore, the change in specific heat cannot explain the suppressed κ in defective MoS2. However, the differences in phonon spectrum such as flattened dispersion usually have a significant influence on phonon transport.84 In particular, the defect-induced quasilocalized phonon states play a dominant role in the significant reduction in thermal conductivity of defective MoS2. The flattened dispersions in phonon spectrum correspond to a few additional small peaks at low frequencies in the phonon density of states (DOS) as marked by orange arrows in Figure 4a, corresponding to quasi-local vibrations.85 These quasilocalized phonon states lead to zero group velocities in Figure 4b, and reductions in the group velocities near the quasilocalized modes. Phonons with small group velocities are not effective carriers of heat,86 leading to low diffusion from 40 to 230 cm−1 compared to perfect MoS2. The flattened dispersions due to quasi-localized states significantly increase the number of three-phonon scattering channels. The three-phonon interaction processes are constrained to satisfy

by frequency, which has been used in previous theoretical calculations.77 For perfect MoS2, the thermal conductivity agrees very well with previous calculations using the same method.52−55,78 The calculated κ for defective MoS2 falls in the range of previous experimental results (30−101 W/mK).45−49 Considering that the point defects are particularly noticeable in monolayer MoS2 due to the imperfection of the growth process,15−19 our predicted values give a reasonable explanation why the measured κ differ significantly. It can be seen that the κ of defective MoS2 is significantly lower than perfect MoS2. The low κ in defective MoS2 with respect to perfect MoS2 is better understood from the analysis of frequency-dependent thermal conductivity, as shown in Figure 2b. The thermal conductivity of defective MoS2 is severely reduced below 200 cm−1. For VS, VS2, and 2VS, 75% of the heat is conducted by phonons with frequencies lower than 70, 50, and 60 cm−1, respectively, indicating that the thermal conductivity is dominated by low frequency phonons. To understand the underlying mechanism of suppressed κ, we compare the phonon dispersion (Figure 3) of all four types of MoS2 in a 3 × 3 × 1 supercell. (The movies of vibrational motions of the phonon states in perfect and defective MoS2 are included in Supporting Information.) The ZA branch in the long-wavelength limit shows quadratic behavior due to low lattice dimensionality.79,80 Different from the band structure of a unit cell,81 the phonon dispersion is triple folded.82,83 Thus, the phonon dispersion of perfect MoS2 seems to be similar to that of defective MoS2, and only slight difference is observed: In defective MoS2, the defects play the role of perturbations in atomic mass and interatomic bonding (see the mass ratio and

ωj ± ωj ′ = ωj ″

(3)

q ± q′ = q″ + K

(4)

where j is phonon branch index, q wave vector, and K is reciprocal-lattice vector. The number of three-phonon scattering channels is usually chracterized by the phase space available for three-phonon processes P3,63,87,88 P3 =

2 ⎛⎜ (+) 1 (−)⎞⎟ P3 + P3 ⎠ 3Ω ⎝ 2

(5)

where Ω is a normalization factor88 and

Figure 2. Calculated thermal conductivity for perfect and defective MoS2 as a function of (a) temperature and (b) phonon frequency. 29326


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Figure 3. Phonon dispersion for (a) perfect MoS2, (b) monosulfur vacancy, (c) disulfur vacancy, and (d) double monosulfur vacancy in a 3 × 3 × 1 supercell.

Table 2. Calculated Defect-to-Perfect Mass Ratio m/M, Largest Harmonic Interatomic Force Constants Φαβ, Specific Heat Capacity Cv, and Three-Phonon Phase Space P3 at 300 K for Perfect and Defective MoS2 perfect VS VS2 2VS

P3(±) =

m/M

Φαβ (eV/Å2)

CV (J/K mol)

P3

1.000 0.978 0.955 0.955

26.15 26.05 25.81 25.63

21.10 20.46 19.81 19.79

0.3 20.3 19.6 19.0

To obtain more insight, we further compare the Normal and Umklapp scattering phase space of perfect and defective MoS2 in Figure 5. As clearly shown in Umklapp scattering phase space of defective MoS2, there is an obvious peak for phonons around 50 cm−1. This can be attributed to the flat modes at about 50 cm−1. The flat modes in defective MoS2 have same ω″λ with different wave vectors qflat = q″ + K over more than one Brillouin zone, therefore there is almost always a mode (ω″λ , qflat) in the quasi-localized region for a phonon mode (ωλ, q) to get scattered, which means that, when a flat mode gets involved in a three-phonon process, the strict requirement of quasimomentum conservation can be relaxed. It should also be noticed that a flat mode can be scattered across the Brillouin zone boundary more easily. Therefore, phonons with frequencies around 50 cm−1 contribute much less to the total thermal conductivity in defective MoS2 than in perfect MoS2 because of much stronger Umklapp scattering and lower diffusion as we mentioned above. Finally, we present three-phonon relaxation time of perfect and defective MoS2 in Figure 6a. Because of increased scattering channels, the relaxation times of defective MoS2 from 50 to 150 cm−1 are much shorter than those of perfect MoS2. In addition to the number of scattering channels, threephonon relaxation time depends on the strength of each scattering channel as well, which is described by the Grüneisen parameter. The mode Grüneisen parameters reflect the anharmonicity of a phonon mode. As shown in Figure 6b,

∑ ∫ dq dq′ δ(ωj(q) ± ωj ′(q′) j,j′,j″

− ωj ″(q ± q′ − K))

(6)

P3 can be further decomposited into Normal and Umklapp scattering, where Normal processes within the first Brillouin zone correspond to K = 0, while Umklapp processes correspond to K ≠ 0. From eqs 5 and 6, we know that the scattering phase space is directly determined from the phonon dispersion. As shown in Table 2, although only a small portion of the phonon branches shift slightly upward and downward to become flattened, the total phase space of perfect MoS2 is much smaller than defective MoS2. The larger number of available scattering channels in defective MoS2 leads to severely reduced lattice thermal conductivity.

Figure 4. (a) Phonon DOS and (b) group velocities for perfect and defective MoS2. 29327


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Figure 5. Three-phonon phase space decomposed into (a) Normal and (b) Umklapp scattering for perfect and defective MoS2.

Figure 6. (a) Total relaxation time and (b) mode Grüneisen parameters for perfect and defective MoS2.

can be used to assess the defect concentrations in semiconductors quantitatively as well.

the Grüneisen parameters for perfect and defective MoS2 are similar to each other. Thus, the increasing scattering channels at low frequencies have to be the governing factor that leads to the reductions in the relaxation time of defective MoS2 from 50 to 150 cm−1.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b10812. Captions for the movies and Figure S1, showing phonon dispersion (PDF) Movie S1: Vibrational motions of the phonon states at 126.2 cm−1 (point A in Figure S1) in the xy plane for perfect MoS2 (AVI) Movie S2: Vibrational motions of the phonon states at 126.2 cm−1 (point A in Figure S1) along the z direction for perfect MoS2 (AVI) Movie S3: Vibrational motions of the phonon states at 115.3 cm−1 (point B in Figure S1) in the xy plane for MoS2 with monosulfur vacancy (AVI) Movie S4: Vibrational motions of the phonon states at 122.6 cm−1 (point C in Figure S1) along the z direction for MoS2 with monosulfur vacancy (AVI)

CONCLUSION

In summary, we explore the thermal transport properties of perfect and defective MoS2 with three types of defects. The thermal conductivity of defective MoS2 (38.2 W/mK-62.1 W/ mK) is much lower than perfect MoS2 due to defective-induced quasi-localized phonon states. These quasi-localized modes lead to low phonon group velocities and increased three-phonon scattering channels, especially for Umklapp scattering by phonons around 50 cm−1 (although the scattering strength of each modes remains similar). Thus, the thermal conductivity of defective MoS2 is severely suppressed. Our results provide a fundamental understanding of the effect of vacancies on thermal transport, and shed light on the manipulation of thermal transport of MoS2 through defect engineering, which 29328


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Movie S5: Vibrational motions of the phonon states at 115.3 cm−1 (point D in Figure S1) in the xy plane for MoS2 with disulfur vacancy (AVI) Movie S6: Vibrational motions of the phonon states at 121.4 cm−1 (point E in Figure S1) along the z direction for MoS2 with disulfur vacancy (AVI) Movie S7: Vibrational motions of the phonon states at 120.1 cm−1 (point F in Figure S1) in the xy plane for MoS2 with double monosulfur vacancy (AVI) Movie S8: Vibrational motions of the phonon states at 120.1 cm−1 (point F in Figure S1) along the z direction for MoS2 with double monosulfur vacancy (AVI)

AUTHOR INFORMATION

Corresponding Authors

*(H.Z.) E-mail: [email protected]. *(HY.Z.) E-mail: [email protected]. ORCID

Hao Zhang: 0000-0002-8201-3272 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China under Grants No. 11374063 and 11404348 and the National Basic Research Program of China (973 Program) under Grant No. 2013CBA01505.

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