First-Principles Study of Manipulating the Phonon Transport of

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First-Principles Study of Manipulating the Phonon Transport of Molybdenum Disulfide by Na-Intercalating Hezhu Shao, Min Jin, Bo Peng, Hao Zhang, Xiaojian Tan, Guoqiang Liu, Haochuan Jiang, and Jun Jiang J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12330 • Publication Date (Web): 18 Dec 2017 Downloaded from http://pubs.acs.org on December 20, 2017

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The Journal of Physical Chemistry

First-principles study of manipulating the phonon transport of molybdenum disulfide by Na-intercalating Hezhu Shao,∗,† Min Jin,† Bo Peng,‡ Hao Zhang,∗,‡ Xiaojian Tan,† Guo-Qiang Liu,† Haochuan Jiang,† and Jun Jiang∗,† Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China., and Shanghai Ultra-precision Optical Manufacturing Engineering Center, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China E-mail: [email protected]; [email protected]; [email protected]

∗ To

whom correspondence should be addressed Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, Chi-

† Ningbo

na. ‡ Shanghai Ultra-precision Optical Manufacturing Engineering Center, Department of Optical Science and Engineering, Fudan University, Shanghai 200433, China

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Abstract Structural modification is an important way to manipulate the phonon transport in materials. Here, we demonstrate that Na-intercalating leads to slightly structural modification, but remarkably reduction in the lattice thermal conductivity for MoS2 . Such reduction is ascribed to the modification of phonon spectra. By Na-intercalating, the phonon dispersion of MoS2 has two remarkable changes. One is the reduction of phonon frequencies, the other is the increased several quasi-local modes of vibration in the range of moderate frequencies. The reduction of the phonon frequencies enhances largely the anharmonicity of low-frequency phonon-phonon interaction and reduces the phonon lifetime by 1∼2 orders of magnitude. And Na-intercalating introduces several quasi-local modes of vibration, which increases the scattering channel of phonon-phonon interaction. We further derive a relation of integrated JDOS and the lattice thermal conductivity, and give a predicable model for the manipulation of phonon transport of MoS2 .

Introduction Recently, transition-metal dichalcogenides (e.g. MoS2 , WS2 , TiS2 , TaS2 ) have been extensively studied and applied in catalysis, energy storage and transform, and electronics. 1–4 Transition-metal dichalcogenides bulk material are layered structures with strong covalent bonding within each layer and weak van der Waals interaction between the layers. The layered structural property of transition-metal dichalcogenides is helpful for inter-layer engineering to control their performance in rechargeable batteries, pseudocapacitors, hydrogen evolution reaction catalysis and treatments of environmental contaminants. 5 Many intercalates, such as organic molecules, alkali metals, and transition-metal halides can be introduced into the inter-layer space. In previous studies, the intercalates have been shown to tailor the physical and chemical properties of layered graphite, hBN, and transition-metal dichalcogenides for certain applications. 6 The intercalates can also help to exfoliate the layered compounds into single or few layers structures. 7 For the applications of transition-metal dichalcogenides, the heat transport and manage is crucial to the performance of 2


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devises. Efficient heat dissipation is favored in high-performance electronic devices, while low lattice thermal conductivity is preferred in thermoelectric applications. Recently, there have been several works to discuss the effect on tuning the thermal conductivity of layered materials by intercalation. For example, Cho et al. found that the thermal conductivity of cobalt oxide could be tuned electrochemically by Li ion intercalation, nevertheless, the mechanism of the electrochemically induced thermal conductivity change was not yet clear in their work. 8 Wan et al. found that after intercalated by organic ions, the thermal conductivity of TiS2 behaved one order of magnitude lower than the single-layer and bulk TiS2 , and their molecular dynamics simulations showed that the acoustic phonons in intercalated TiS2 had less relaxation time that in pristine TiS2 . 9 Qian et al. found the anisotropic tuning of graphite thermal conductivity by intercalation of Li with different concentration through molecular dynamics simulations, and in their work, the quasi-local vibrational modes related to Li atom were considered as the main source of reduction of thermal conductivity along in-plane direction by Li intercalation. 10 By molecular dynamics simulations, Dunn found Cu intercalation could tune the thermal conductivity of MoTe2 . 11 As for MoS2 , Zhu et al. found that Li intercalation into MoS2 by electrochemical method could modified the thermal conductivity, and they further showed the ration of the in-plane to across plane thermal conductivity were enhanced by Li intercalation. 12 Inspired by previous work of intercalation induced change of thermal conductivity in layered materials and to understand the mechanism of intercalating on the phonon transport in MoS2 , we employed first-principles calculations to study the phonon transport property for MoS2 before and after Na-intercalating. In this work, we demonstrate two key factors, one is structural changing induced changed phonon dispersion and the other is the quasi-local vibration introduced by the impurity atom, for manipulating the phonon transport in MoS2 by Na-intercalating. Often the intercalates could influence dramatically the structural properties between layers, while slightly within layers. Then it is believed that the properties related to the in-plane structure are slightly modified by the intercalates. While we find that Na-intercalating could make more influence on the in-plane phonon transport of MoS2 than on that along inter-layer (out-of-plane) direction. By elaborate analyzing

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the modifications of structure, elastic constants, lattice dynamics and thermodynamics, phonon lifetime, anharmonicity, and phonon-phonon scattering channel of MoS2 by Na-intercalating, we have investigated in-depth the effect of Na-intercalating on the phonon transport of MoS2 . This provides an insight to manipulating the phonon transport in similar materials.

Methodology The calculations are based on density functional theory method in the generalized gradient approximation with the Perdew, Burke, and Ernzerhof functional (PBE), 13 as implemented in the Vienna Ab initio Simulation Package (VASP), which employs a plane-wave basis. 14,15 During relaxations, the plane-wave energy cutoff is set to be 350.00 eV, the electronic energy convergence is 10−5 eV, and the force convergence for ions is 10−3 eV/Å. The van der Waals correction DFT-D2 method of Grimme 16 is employed to correctly describe the inter-layer properties of MoS2 and Na-intercalated MoS2 . Parlinski-Li-Kawazoe method, which is based on the supercell approach with finite displacement method as implemented in the Phonopy package, 17,18 is employed to obtain the phonon dispersion, group velocities, and thermodynamical properties. To obtain convergent lattice dynamical properties, in the calculations of harmonic interatomic force constants, a 4 × 4 × 2 supercell of primitive cell containing 192 atoms for MoS2 and 224 atoms for Na-intercalated MoS2 is employed, and a Γ–centered 3 × 3 × 2 Monkhorst-Pack k-point mesh is used to sample the irreducible Brillouin zone. The phonon Boltzmann transport equation solved iteratively as implemented in ShengBTE 19 is employed to obtain the phonon lifetime. The lattice thermal conductivity κL is calculated by αβ

κL =

1 β f0 (ωλ )( f0 (ωλ ) + 1)(¯hωλ )2 vαλ vλ τλ . ∑ 2 NkB T Ω λ

(1)

where Ω is the volume of unit cell, α and β are the Cartesian components of x, y, or z, kB is Boltzmann constant, and f0 (ωλ ) is Bose-Einstein distribution function. To get the three-phonon 4


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scattering matrix elements for calculating the phonon lifetimes, the third-order force constants should be calculated firstly. In ShengBTE, the third-order force constants are calculated by a finitedifference supercell approach. For the force calculations, a 4 × 4 × 2 supercells of primitive cell for both MoS2 and Na-intercalated MoS2 are employed so that there is only negligible interaction between atoms in the center and at the boundary. To get convergent lattice thermal conductivity, we adopt 5.5 and 5.59 Å as the cutoff of third-order force interactions for MoS2 and Na-intercalated MoS2 , respectively. And there are 320 and 744 supercells with displaced atoms needed to be calculated in VASP for MoS2 and Na-intercalated MoS2 , respectively. During the calculations of force constants, only the Γ point is set for the k-point grid. In addition, a parameter, scalebroad in ShengBTE, should be set manually. Here, we test several scalebroad values of 0.1, 0.5, and 1.0, and find 0.5 is enough to obtain the convergent lattice thermal conductivities. And in the calculation of lattice thermal conductivity, a 29 × 29 ×7 q-point grid (340 inequivalent q points) is employed to reach convergence.

Results and Discussion

Figure 1: The primitive cell of (a) MoS2 and (b) Na-intercalate MoS2 . Bulk MoS2 has layered hexagonal structure with space group of P63 /mmc (194). We here consider a situation of Nax MoS2 (x = 0.5), the Na atom are intercalated between two layers of MoS2 as ¯ shown in Fig. 1. By intercalated of Na, the symmetry turns into P31m (164). The point group determines that the symmetry operations in MoS2 is twice as much as those in Na-intercalated MoS2 . 5



(a)

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(b)

Figure 2: Temperature-dependent thermal conductivities of MoS2 (a), where points are experimental data extracted from Ref., 12 and Na-intercalated MoS2 (b), along in-plane and out-of-plane directions. The inset figure in (b) shows the ratio of lattice thermal conductivities of Na-intercalated MoS2 to those of MoS2 . Table 1: Elastic properties of MoS2 and Na-intercalated MoS2 . including the elastic constants ci j (in GPa), the isotropic bulk modulus (B), shear modulus (G), and Young’s modulus (E in GPa) for polycrystalline obtained from the single crystal elastic constants using Hill’s approximations, and percentage anisotropy in compressibility (AB ) and shear (AG ) moduli.

MoS2 Exp. Na:MoS2

AB AG c33 c44 c12 c13 c14 BV BR GV GR E 49.9 15.1 57.0 10.3 69.9 40.3 47.9 26.3 90.9 26.8% 29.2% 52∗ 158.3 78.7 22.5 28.5 29.2 5.0 63.2 57.2 42.6 33.2 93.9 5.0% 12.4% c11 211.9

*from Ref. 12 The unit cell of MoS2 contains two groups of S-Mo-S unit, and Mo and S atoms occupies two-fold position 2c(1/3,2/3,1/4) and four-fold position 4f(1/3,2/3,z(S)), respectively. After Na intercalating, the S atoms are divided into two two-fold positions 2d(1/3,2/3,z(S1 )) and 2d(2/3,1/3,z(S2 )). To obtain the relaxed structure, firstly we chose several volumes around the expected equilibrium volume and relaxed them for ions and shapes, and got a set of volumes Vi and energies Ei , then fit these (Vi , Ei )s to the Birch-Murnaghan 3rd-order equation of state. 20 The obtained lattice constants of a and c of MoS2 are 3.19 and 12.43 Å, respectively, which are slightly larger than the experimental data of a = 3.15 and c = 12.3 Å. 21 The calculations by PBE with van der Waals correction overestimates a and c by only 1.0% and 1.3%, respectively. It shows the calculations are in good agreement with experiments. After intercalated by Na, the volume of MoS2 expand both along c (out-of-plane) direction and the direction perpendicular to c (in-plane). Compared with those in MoS2 , the lattice constants of Na-intercalated (a = 3.23 and c = 13.66 Å) MoS2 enlarge by 1.2% 6


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and 9.9% for a and c respectively, indicating that Na-intercalating influences more remarkably the structure between layers than that within layers. To investigate the effect of intercalating by Na on the phonon transport of MoS2 , we have compared the lattice thermal conductivities for MoS2 and Na-intercalated MoS2 (Fig. 2). Due to the layered structural character, both MoS2 and Na-intercalated MoS2 behave strong anisotropic in phonon transport. And the lattice thermal conductivities along in-plane and out-of-plane directions comply well with 1/T relation in a wide range of temperature (200∼800 K) before and after Naintercalating in MoS2 . Our calculated lattice thermal conductivities along in-plane and out-ofplane directions at room temperature for bulk MoS2 are 116.0 and 6.7 Wm−1 K−1 respectively, which are both slightly greater than the experimental values (107 and 2 Wm−1 K−1 ). 12 Compared with experimental value, the in-plane lattice thermal conductivity has relative error within 9%, and the out-of-plane one deviates relatively larger, while is in the same order of magnitude. The discrepancy between the present calculations and the experimental measurements may come from three respects. Firstly, the PBE calculations overestimate valence bond length of Mo-S, which leads to underestimated interatomic interactions. Secondly, during calculations, only three-phonon interaction is considered here, and higher order phonon-phonon interactions such as four-phonons interactions which bring more scattering are neglected. Lastly, in experiments, the samples of bulk MoS2 are somewhat contain some defects, which could lead to the reduction of lattice thermal conductivities. As expected, Na-intercalating results into dramatic influence on the out-of-plane phonon transport of MoS2 , owing to the structural transformation by inserted Na atom. From Fig. 2, Naintercalating leads to decreasing of out-of-plane lattice thermal conductivity by around 70%. On the other hand, the Na-intercalating causes more substantial decreasing (by over 80%) for the in-plane lattice thermal conductivities than out-of-plane ones. It contradicts seemingly with that intercalates could influence more dramatically the properties inter-layers than those within layers. However, by introducing Coulomb interaction between Na ions and negatively charged MoS2 layers, the Na-intercalating can bring forth stronger attraction between the layers of MoS2 than that

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by mere van der Waals interaction. The structural change of MoS2 by Na-intercalating can partly reflects above reasoning. The Na-intercalating slightly enlarges the valence length of S-Mo by 0.25%, and more importantly poses some stress to the S-Mo-S layers, leading to a smaller valence angle of S-Mo-S in Na-intercalated MoS2 , as shown in Fig. 1. Additionally, the distance between S-Mo-S layers in MoS2 is 3.1 Å, while after Na-intercalating, the distance between the unit of Na(MoS2 )2 is reduced to 3.06 Å, which also demonstrates the effect of compression to the MoS2 layers by Na-intercalating. Then the degree of anisotropy in phonon transport would reduce, which brings that more relative decrease of in-plane lattice thermal conductivities than out-of-plane ones. The anisotropy related to heat transport of (Na-intercalated) MoS2 can be estimated further by the elastic property. The structure of MoS2 belongs to the hexagonal system, which contains five independent elastic constants. After intercalated by Na atom as shown in Fig. 1(b), the structure transfers into the trigonal system, which has six independent elastic constants. As shown in Table 1, the calculated elastic constant of c33 of MoS2 is well consistent with experimental value. 12 The elastic constants in Table 1 satisfy all the Born’s mechanically stability criteria, 22 which indicates that both MoS2 and Na-intercalated MoS2 are in mechanical stable states. Using the calculated elastic constants, the bulk and Young’s modulus can be obtained by VoigtReuss-Hill approximations. 23 And from the Voigt-Reuss bulk and Young’s modulus, we can calculate the percentage anisotropy in compressibility and shear, AB = (BV − BR )/(BV + BR ), and AG = (GV − GR )/(GV + GR ), which are introduced by Chung and Buessem. 24 The two values AB and AG range from zero, which represents perfect isotropy, to 100%, which represents the largest possible anisotropy. The calculated elastic properties clearly show that after intercalated by Na atom in MoS2 , a considerable reduction in percentage anisotropy of bulk (by more than 4 times) and shear moduli (from 29.2% to 12.4%). On the other hand, the elastic anisotropy can reflected by the deviation of ratios of c11 /c33 from 1. The c11 /c33 of MoS2 behaves greater deviation from 1 than that of Na-intercalated MoS2 . Therefore, it is obvious that the degree of anisotropy in elastic property decreases by Na atom intercalated in the MoS2 , so does the degree of anisotropy of phonon transport.

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 3: Phonon spectra of MoS2 (a) and Na-intercalated MoS2 (b). The red circle points in (a) are experimental data by neutron scattering measurements in Ref., 21 and blue square points are from Raman data in Ref. 25 The different colors in the phonon spectra depict different phonon branches. And their corresponding group velocities along Γ–M and Γ–A, where (c) is for MoS2 and (d) is for Na-intercalated MoS2 . (e), (f), and (g) are for atoms Na-, Mo-, and S-projected phonon spectra for Na-intercalated MoS2 respectively.

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Our prime interest lies in understanding how does Na-intercalating bring such dramatic decreasing to the phonon transport of MoS2 . The key physical quantities related to the phonon transport such as heat capacity (Cv ), Debye temperature (ΘD ), Grüneisen parameter (γ ), channels for phonon-phonon scattering events and so on are decided by the phonon dispersion. Firstly, we analyse the phonon spectrum of bulk MoS2 , shown in Fig. 3(a). We also depicted the experimental data by neutron inelastic scattering and Raman scattering for comparing. The overall agreement between our calculations and experiments is good. This indicates that the PBE with van der Waals corrections describes reasonably well such layered systems with week inter-layer interaction. The character of phonon vibrational properties of bulk and few-layer MoS2 has been extensively studied. 26,27 Here we will focus on the effect of Na-intercalating on the phonon spectra of MoS2 . The phonon spectrum of Na-intercalated MoS2 shown in Fig. 3(b) has no imaginary frequency, indicating a stable phase. It is consistent with above elastic analysis. This prediction makes for the preparation of Na-intercalated MoS2 . The enlarged valence length (by mere 0.25%) of S-Mo by Na-intercalating brings the second-order interatomic force constants of S-S and Mo-Mo decreasing by 19% and 10% respectively. This leads to an overall diminishing for the vibrational frequencies. For instance, compared with MoS2 , the highest vibrational frequency of Na-intercalated MoS2 dropped by around 10%. And Na-intercalating results in even larger decrease for the low-frequency modes below the gap, which results in a larger gap (41.2 cm−1 for Na-intercalated MoS2 and 37.5 cm−1 for MoS2 ) separating the high-frequency optical branches from the low-frequency ones. In general, the reduction in phonon frequencies could bring about decreasing in group velocities and increasing phonon-phonon scattering, and then lead to reduction in lattice thermal conductivity. As shown in Fig. 3(c) and Fig. 3(d), along Γ–M near zone center, the group velocities (GVs) of longitudinal acoustic (LA) and transverse acoustic (TA) branches of Na-intercalated MoS2 are lower than those of MoS2 by 4% (6.6 km/s for MoS2 and 6.3 km/s for Na-intercalated MoS2 ) and 8% (4.0 km/s for MoS2 and 3.7 km/s for Na-intercalated MoS2 ), respectively. While for the out-of-plane acoustic (ZA) along Γ–M, the GV (2.6 km/s) of Na-intercalated MoS2 is larger than that (2.0 km/s) of MoS2 . Along Γ–A, the GVs of LA (4.1 km/s) and TA (2.3 km/s) modes of Na-intercalated MoS2

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are both larger than those of MoS2 (3.2 km/s for LA mode and 1.8 km/s for TA mode). This reflects the stronger cohesion in out-of-plane direction in Na-intercalated MoS2 , which is consistent with above discussion of structural and elastic properties. We will further analyze the strengthening of phonon-phonon scattering owing to the reduction in phonon frequencies in later paragraphs. In addition, as shown in Fig. 3(b), Na-intercalating introduces two extra optical modes with frequencies of 123.4 and 220.6 cm−1 at Γ point. The former mode with 123.4 cm−1 displays some dispersion along in-plane, for instance Γ-M, direction, and remains flat along out-of-plane (Γ-A) direction. While the higher one exhibits almost local vibrating all over the Brillouin zone. Fig. 3(e) shows these two extra optical modes are contributed mainly by Na vibration. We will discuss further the effect of these modes on the decrease of the lattice thermal conductivity of Na-intercalated MoS2 .

Figure 4: Symmetry and normal displacements of vibration modes for Na-intercalated MoS2 . Next we analyze the vibrational properties of Na-intercalated MoS2 . At the Γ point, the lattice vibrations can be classified according to the symmetry group of crystals. The structure of Naintercalated MoS2 belongs to the D3 d point group and its unit cell contains two S-Mo-S units and a Na atom in between the S-Mo-S layers. And there are 21 phonon modes (3 acoustic and 18 optical modes), which can be classified as Γ = 3A1 g + 4A2 u + 4Eu + 3Eg. One A2 u and one Eu are acoustic modes, and A1 g and Eg are Raman (R) active, another A2 u and Eu are infrared (IR) active. All the Eu and Eg modes are doubly degenerate and vibrate in the xy plane. Because of the 11



(a)

(b)

(c)

(d)

Figure 5: Thermodynamical properties of MoS2 and Na-intercalated MoS2 . (a) temperature dependent heat capacity in unit volume Cv , (b) Debye temperature θD above 4 K, Grüneisen parameter γ , and ratio of Slack descriptor of Na-intercalated MoS2 to that of MoS2 . presence of inversion symmetry, these R and IR modes are mutually exclusive in Na-intercalated MoS2 . We depict the vibrational modes based on above group theory analysis and respective characteristic of vibrations in Fig. 4. We note here that the Na atom related vibrational modes with 123.4 (Eu mode) and 220.6 cm−1 (A2 u mode) are both IR active. The interatomic force constants of Na-Na along zz are larger than that along xx by around 60%, which leads to the larger frequency of out-of-plane A2 u mode than that of in-plane Eu mode. We hope above analysis can facilitate future experimental measurement for Na-intercalated MoS2 . Statistically physical quantities related to the lattice dynamics such as heat capacity, Debye temperature, Grüneisen parameter and so on, which are determined by phonon dispersion relation, can used to appraise the phonon transport and be helpful to the understanding of mechanisms for the change of lattice thermal conductivities. According to phonon gas model, the lattice thermal conductivity is determined by three factors. κL = 31 CvV 2 τ , where Cv , V , and τ are heat capacity in unit volume, acoustic velocity, and phonon lifetime, respectively. The phonon lifetime is decided by the phonon-phonon interactions. The strength of phonon-phonon interactions is closely related to the anharmonicity, described by Grüneisen parameter. Slack developed above model and 12


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Mθ 3 δ

D proposed a formula to estimate the lattice thermal conductivity. 28 κL = A γ 2 n2/3 , where θD is the T

Debye temperature and γ is the Grüneisen parameter. We could draw θD3 /γ 2 as a descriptor, named Slack descriptor for convenience, to estimate the relative level of phonon transport. (a)

(b)

(c)

Figure 6: Frequency dependent (a) phonon lifetime τ , (b) Grüneisen parameters γ (ω ), and (c) joint density of state for the scattering rate of MoS2 and Na-intercalated MoS2 . In Fig. 5, we compare several important thermodynamical properties of MoS2 and Na-intercalated MoS2 , such as Cv , θD , γ , and the ratio of Slack descriptor of Na-intercalated MoS2 to that of MoS2 . The Cv s are slightly elevated throughout all the temperature range by Na-intercalating (Fig. 5(a) ), due to more vibration modes introduced regardless of the expanded volume. It is favor for the heat transport. From the Cv , the Debye temperatures θD are obtained by fitting the Debye formula in Fig. 5(b). The θD of MoS2 is a monotonically increasing function with respect to temperature above 8 K. This behavior is common in those materials with extremely weak inter-layer interactions. 21 Whereas by Na-intercalating, the θD firstly fall and then rise with the increasing of temperature. The turning point is about 26 K. This behavior of temperature dependent θD in Na-intercalated MoS2 is more similar to normal materials having moderate or slight anisotropy. It reflects that Naintercalating weakens the anisotropy of MoS2 , which is consistent with above discussion. Above 21 K, Na-intercalating makes the θD decrease, which weaken the heat transport according to Slack formula. Fig. 5(c) shows that the Grüneisen parameters of MoS2 are smaller than those of Na-intercalated MoS2 throughout all the temperature range. This implies that Na-intercalating strengthens the anharmonicity of MoS2 , which leads to reduction in phonon lifetime. Fig. 5(d) gives the temperature dependent ratio (D p ) of Slack descriptor of Na-intercalated MoS2 to that of MoS2 . With increasing temperature, D p rises dramatically at low temperature range and saturates to around 0.5 at high temperature. This behavior shows that at low temperature, Na-intercalating 13



could induce more dramatic decrease to the lattice thermal conductivity of MoS2 , despite of underestimating the degree of decreasing in thermal conductivity, which drops by about 70% and 80% for out-of-plane and in-plane values shown in Fig. 2(b). This prediction is consistent with the calculation of in-plane thermal conductivities in the inset of Fig. 2(b). Such behavior of D p is closely related to the changing of Grüneisen parameters as shown in Fig. 5(c). Because only low-frequency phonon modes are activated at low temperature, we speculate that Na-intercalating strengthens more anharmonicity of low-frequency phonon modes, and has lesser influence on the anharmonicity of high-frequency phonon modes. To gain deeper physical insight into the phonon transport of MoS2 by Na-intercalating, we have compared the frequency dependent phonon lifetime τ , Grüneisen parameters, joint density of state (JDOS) for the scattering rate, of MoS2 and Na-intercalated MoS2 in Fig. 6. There is a considerable reduction in phonon lifetime after Na intercalating for all frequencies, and especially in the frequency range of 100∼200 cm−1 , the phonon lifetimes of Na-intercalated MoS2 are lower than those of MoS2 by two order of magnitude. This implies that at the frequency range of 100∼200 cm−1 , the phonon-phonon scattering are enhanced remarkably by Na-intercalating. The phonon-phonon scattering is determined by two factors: anharmonicity and scattering channel for phonon-phonon interactions. Fig. 6 (b) shows that the frequency dependent Grüneisen parameters of Na-intercalated MoS2 in high frequency range behave similar to those of MoS2 , whereas in the range of 100∼200 cm−1 , behave much larger value than those of MoS2 . This indicates that the Na-intercalating strengthens the anharmonicity of relative low-frequency phonon modes, while has almost no influence on the anharmonicity of high-frequency phonon modes. It is consistent with above discussion for the behavior of temperature dependent Grüneisen parameter. Fig. 6 (c) shows the joint density of state (JDOS) for the phonon scattering rate. 29 In the low frequency range lower than 100 cm−1 , the JDOS of MoS2 and Na-intercalated MoS2 are very close to each other, which indicates Na-intercalating has no effect on the phonon-phonon scattering channels in the low-frequency range. From 100 to 220 cm−1 , where near the characteristic frequencies of Na vibration, the JDOS of Na-intercalated MoS2 has considerable enhancement comparing with

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that of MoS2 . And in the high-frequency optical modes near the characteristic frequencies of Na vibration, the JDOS of Na-intercalated MoS2 enhances as well. This behavior demonstrates the importance influence by Na-intercalating is to provide more phonon-phonon scattering pathes, and then leads to dramatic decrease for the phonon lifetime at certain frequency range. (a)

(c)

(b)

(d)

(e)

′

Figure 7: Phonon spectra of (a) Na-intercalated MoS2 removing the Na vibration (MoS2 ) , and (b) of Na-intercalated MoS2 vs. that of MoS2 . Frequency dependent (c) phonon lifetime τ , (d) ′ Grüneisen parameters γ (ω ), and (e) JDOS for the scattering rate of (MoS2 ) and Na-intercalated MoS2 . To find out what exactly the effect of changed JDOS owing to Na vibration on the reduction of lattice thermal conductivity, we then discriminate the factor of changed JDOS from that by reduction in phonon frequencies bringing larger anharnicity and lower GVs. In Fig. 7 (a) and (b), we give the phonon spectra of Na-intercalated MoS2 removing the Na vibration, here named ′

(MoS2 ) , and Na-intercalated MoS2 vs. that of MoS2 . Compared with the phonon dispersion of ′

MoS2 , the reduction in frequency for (MoS2 ) originates from the structural transformation by Na′

intercalating. The phonon lifetimes of (MoS2 ) are slightly larger than those of Na-intercalated ′

MoS2 , however, have the same order of magnitude with the later (Fig. 7 (c)). And (MoS2 ) and Na-intercalated MoS2 have almost the same Grüneisen parameter (Fig. 7 (d)), which indicates that they exhibit similar anharmonicity. It means that the reduction in frequency of MoS2 by Na-intercalating strengthens the anharmonicity of MoS2 and leads to considerable reduction in 15



phonon lifetime. Further introducing Na vibration in phonon spectra have almost no influence on the enhancing of anharmonicity. On the other hand, the Na vibration introduces more scattering channels for phonon-phonon interaction (Fig. 7 (e)), which also help for reducing the phonon ′

lifetimes (Fig. 7 (c)). For (MoS2 ) , we obtain the in-plane lattice thermal conductivity of 35.8 Wm−1 K−1 at room temperature. Compared with that of MoS2 , the lattice thermal conductivity ′

of (MoS2 ) decreases by 69%. The enhanced JDOS by Na vibration further lead to reduction of lattice thermal conductivity by 39% (21.8 Wm−1 K−1 of Na-intercalated MoS2 vs. 35.8 Wm−1 K−1 ′

of (MoS2 ) ). Then this demonstrate that the quasi-local modes induce large reduction of lattice thermal conductivity of MoS2 , while the reduction of frequencies by structural transformation by Na intercalating leads to more considerable decrease for the lattice thermal conductivity.

Figure 8: The relation of integrated JDOS and lattice thermal conductivity in Na-intercalated MoS2 . To make predictable the effect of quasi-local modes introduced by Na vibration on lattice thermal conductivity, we study the relation between the integrated JDOS (P3 in Fig. 8) and lattice thermal conductivity. We construct the systems with Li and K substitute for Na atom without changing the structure of Na-intercalated MoS2 . Here, we also keep the force constants of systems containing Li and K atoms to be identical to those of Na-intercalated MoS2 . Owing to the different atomic mass from Na atom, the Li and K vibrations will introduce different vibrational modes into the phonon spectra of MoS2 , and lead to different JDOS. Specially, K has larger atomic mass than Na atom, the frequencies of vibrational modes introduced by K are lower than those by Na atom. And Li has the opposite situation. From the lattice thermal conductivity and P3 of systems con′

taining Li, Na, K, and (MoS2 ) , we obtain a fitting relation P3 = 3.97 × 10−2 κL−0.79 in Fig. 8. The 16


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exponent −0.79 could be used to estimate the difficulty of adjusting the lattice thermal conductivity of MoS2 by changing the phonon-phonon scattering channel. It should be determined by the specific phonon dispersion of material. This relation describes quantitatively the effect of changed JDOS on the phonon transport in Na-intercalated MoS2 , could be used to serve for manipulating the heat transport in MoS2 . For instance, from this formula, increasing the P3 by 73% could leads to reduction the lattice thermal conductivity by 50%. We note here that the Na concentration is another important factor in modifying the lattice thermal conductivity of MoS2 . With changing of Na concentration and different structural modification, the phonon dispersion will be modified accordingly. If the Na concentration is lower than that considered above (Nax MoS2 , where x < 0.5), the structural change may induce less reduction of phonon frequencies, while the quasi-local phonon modes would be more localized. Such two effects are competitive with each other in modifying the lattice thermal conductivity. Then the effect of Na concentration on the phonon transport of MoS2 should be further investigated.

Conclusion In summary, the results presented here demonstrate a considerable decrease of lattice thermal conductivity of MoS2 by Na-intercalating. By analysing the structural, lattice dynamical, thermodynamic, and phonon transport properties of MoS2 before and after intercalated by Na, we clarify the mechanism of Na-intercalating induced reduction to the lattice thermal conductivity of MoS2 . Among several key factors influencing the phonon transport, we make a distinction between the quasi-local vibrational modes introduced by Na atoms and other factors such as heat capacity, Debye temperature, anharmonicity and so on. In Na-intercalated MoS2 , the most prominent factor of influencing the phonon transport is the structural change induced reduction of phonon frequencies. In addition, the quasi-local vibrational modes introduced by Na atoms could further decrease the lattice thermal conductivities considerably. The results presented here also suggest an inverted design to enhance the heat transport of MoS2 by increasing the whole phonon frequencies by

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some structural engineering. Our work facilitates deep understanding the mechanism of phonon transport of MoS2 by Na-intercalating and provides an insight into the manipulation of phonon transport in similar materials.

Acknowledgement This work was supported by the National Natural Science Foundation of China (11404348, 11374063, 11404350, and 11234012), Natural Science Foundation of Zhejiang Province (LY17A040012), Public Projects of Zhejiang Province (2017C31006), Zhejiang Provincial Science Fundation for Distinguished Young Scholars (LR16E020001), and Ningbo Municipal Natural Science Foundation (2017A610009 and 2017A610107).

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