Beyond Perturbation: Role of Vacancy-Induced Localized Phonon

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Beyond Perturbation: Role of Vacancy-Induced Localized Phonon States in Thermal Transport of Monolayer MoS 2

Bo Peng, Zeyu Ning, Hao Zhang, Hezhu Shao, Yuanfeng Xu, Gang Ni, and Heyuan Zhu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b10812 • Publication Date (Web): 08 Dec 2016 Downloaded from http://pubs.acs.org on December 11, 2016

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The Journal of Physical Chemistry

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 E-mail: [email protected]; [email protected]

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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 quasi-localized 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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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 ultra-highquality 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, 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 W/mK, 45 62.2 W/mK 46 and 84±17 W/mK 47 for monolayer MoS2 , 52 W/mK 48 and 44-52 W/mK 49 for few-layer MoS2 , and 110±20 W/mK for bulk MoS2 , 50 respectively. The measured κ of monolayer 3



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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 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 quasi-localized 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 . (a)

(b)

(c) VS

(d)

bVS

VS2 aVS

VS

VS1

aVS

bVS

VS2

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.

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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 method 65 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 inplane lattice thermal conductivity is isotropic 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 5



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vector q, κ = καα =

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1 X 2 Cλ vλα τλα , V λ

(1)

where V is the crystal volume, Cλ is the heat capacity per mode, 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 threephonon processes are approximated by a Gaussian function with a scale parameter of 1 for broadening. 64 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) towards κ in perfect and defective MoS2 . Perfect VS VS2 2VS

Eform (eV) κ (W/mK) 154.3 6.69 62.1 13.84 38.2 13.38 40.2

ZA (%) 29.0 11.5 9.7 17.2

TA (%) 30.5 23.1 27.1 22.6

LA (%) Optical (%) 39.1 1.4 47.6 17.8 51.3 11.9 40.9 19.3

Results and discussion The crystal structures of perfect and defective MoS2 are shown in Fig. 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 [Fig. 1(c)], while for 2VS, two S atoms are missing from two sites of the first layer [Fig. 1(d)]. We first investigate the 6


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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 high-resolution transmission electron microscopy experiments, sulfur vacancies are formed under exposure to an electron beam, which is much easier than molybdenum vacancy formation. 23 (a)

(b)

Figure 2: Calculated thermal conductivity for perfect and defective MoS2 as a function of (a) temperature and (b) phonon frequency. The lattice thermal conductivity κ of perfect and defective MoS2 as a function of temperature is shown in Fig. 2(a). 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 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 7



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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 Fig. 2(b). 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 cm−1 , 50 cm−1 and 60 cm−1 , respectively, indicating that the thermal conductivity is dominated by low frequency phonons. (b) VS

(a) Perfect

(c) VS2

(d) 2VS

500 400 300 200 100 0 Γ

M

K

Γ

M

K

Γ

M

K

Γ

M

K

Γ

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. To understand the underlying mechanism of suppressed κ, we compare the phonon dispersion 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 Supplementary Material.) 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 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

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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 quasi-localized phonon states play a dominant role in the significant reduction in thermal conductivity of defective MoS2 . Perfect

VS

VS2

2VS

DOS

(a)

(b) 7 Group velocity (km/s)

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6 5 4 3 2 1 0 0

100

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400

0

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0

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0

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500

Figure 4: (a) Phonon DOS and (b) group velocities for perfect and 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 Fig. 4(a), corresponding to quasi-local vibrations. 85 These quasi-localized phonon states lead to zero group velocities in Fig. 4(b), 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 cm−1 to 230 cm−1 compared to perfect MoS2 . Table 2: Calculated defect-to-perfect mass ratio m/M , largest harmonic interatomic force constants Φαβ , specific heat capacity CV , three-phonon phase space P3 at 300 K for perfect and defective MoS2 . m/M Perfect 1.000 VS 0.978 VS2 0.955 2VS 0.955

Φαβ (eV/Å2 ) CV 26.15 26.05 25.81 25.63

(J/Kmol) P3 21.10 0.3 20.46 20.3 19.81 19.6 19.79 19.0

The flattened dispersions due to quasi-localized states significantly increase the number 9



of three-phonon scattering channels. The three-phonon interaction processes are constrained to satisfy 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 =

1 (−) 2 (+) (P3 + P3 ), 3Ω 2

(5)

where Ω is a normalization factor, 88 and (±) P3

=

X Z

(6)

dqdq′ δ(ωj (q) ± ωj ′ (q′ ) − ωj ′′ (q ± q′ − K)).

j,j ′ ,j ′′

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 K6=0. Perfect

VS

VS2

2VS

Phase space for Normal scattering

(a)

(b) Phase space for Umklapp scattering

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0

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400

0

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0

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0

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500

Figure 5: Three-phonon phase space decomposed into (a) Normal and (b) Umklapp scattering for perfect and defective MoS2 . 10


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From Eq. (5)-(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 upwards and downwards 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. To obtain more insight, we further compare the Normal and Umklapp scattering phase space of perfect and defective MoS2 in Fig. 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 ωλ′′ with different wave vectors qflat = q′′ + K over more than one Brillouin zone, therefore there is almost always a mode (ωλ′′ , 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 quasi-momentum 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. Perfect

Relaxation time (ps)

4 (a) 10

VS2

2VS

2

10

1

10

0

10 10

VS

3

10

10

(b) Grüneisen parameter

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-1 -2

2 1 0 -1 -2 -3 -4 -5 -6

0

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0

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0

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

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Finally, we present three-phonon relaxation time of perfect and defective MoS2 in Fig. 6(a). Due to increased scattering channels, the relaxation times of defective MoS2 from 50 cm−1 to 150 cm−1 are much shorter than those of perfect MoS2 . In addition to the number of scattering channels, three-phonon 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 Fig. 6(b), 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 cm−1 to 150 cm−1 .

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

Acknowledgement 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. 12


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Supporting Information Available • defect-sub-SI This material is available free of charge via the Internet at http://pubs.acs.org/.

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Beyond Perturbation: Role of Vacancy-Induced Localized Phonon

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