Lattice Vibrational Modes and Phonon Thermal Conductivity of Monolayer MoS2
1 Lattice Vibrational Modes and Phonon Thermal Conductivity of Monolayer MoS Yongqing Cai, Jinghua Lan, Gang Zhang* and Yong-Wei Zhang
Institute of High Performance Computing, A*STAR, Singapore 138632 *Email: [email protected]
ABSTRACT:
The anharmonic behavior of phonons and intrinsic thermal conductivity associated with the Umklapp scattering in mono-layer MoS sheet are investigated via first-principles calculations within the framework of density functional perturbation theory. In con-trast to the negative Grรผneissen parameter ( ๐พ ) occurring in low frequency modes in graphene, positive ๐พ in the whole Brillouin zone is demonstrated in monolayer MoS with much larger ๐พ for acoustic modes than that for the optical modes, suggesting that monolayer MoS sheet possesses a positive coefficient of thermal-expansion. The calculated phonon lifetimes of the infrared active modes are 5.50 and 5.72 ps for E โฒ and A respectively, in good agreement with experimental result obtained by fitting the dielectric oscillators with the infrared reflectivity spectrum. The lifetime of Raman A mode (38.36 ps) is about 7 times longer than those of the infrared modes. The dominated phonon mean free path of monolayer MoS is less than 20 nm, about 30-fold smaller than that of graphene. Combined with the non-equilibrium Greenโs function calculations, the room temperature thermal conductivity of monolayer MoS is found to be around 23.2 Wm -1 K -1 , two orders of magnitude lower than that of graphene. I. INTRODUCTION
As a semiconducting analogue of graphene, the atomically thin MoS monolayer, consisting of a hexagonal lattice of Mo atoms sandwiched between two similar lattices of S atoms in a trigonal prismatic arrangement, has recently attracted considerable atten-tion for field effect transistor (FET) and optical device applica-tions due to its presence of a finite bandgap [1, 2]. Great efforts have been made to understand the dynamics of carriers of MoS including mobilities of excitons [3, 4], electrons/holes [5, 6] and the effects from electrical [7] and stress [8-10] fields applied within the layers. In contrast to graphene, dielectric screening associated with the electron-electron interaction and electron-phonon coupling of the inter- and intra-layers gives rise to an anomalous frequency shift [11], a symmetry-dependent phonon renormalization [12], and a superconducting behavior in MoS [13]. In addition to the potential applications in FET devices, MoS has recently shown an intriguing capability of a thermoelectric energy conversion, where a large value of the Seebeck coefficient for single-layer MoS ( โ4 ร 10 and โ1 ร 10 ๐VK โ1 depend-ing on the strength of the external electric field) was demonstrated [14]. For both FET and thermoelectric applications, the phonon property of monolayer or few-layer (FL) MoS is critical. On the one hand, the electron-acoustic phonon coupling dominates the scattering of low-energy carriers, which limits the carrier mobility [5,6]. On the other hand, monolayer MoS is a semiconductor with a sizable bandgap. Thus the electrons have limited contribu-tion to thermal conductivity ( ๐ ), and the intrinsic ๐ is dominated by phonon contribution. Recent molecular dynamics (MD) simu-lations showed that the ๐ value for monolayer MoS is 1.35 Wm -1 K -1 by Liu et al. [15] and 6 Wm -1 K -1 by Jiang et al. [16] Experi-mental measurements of the ๐ of FL MoS were reported to be between 0.4-1.59 Wm -1 K -1 [17,18]. More recently, using Raman spectroscopy approach, Sahoo et al. reported a value of around 52 Wm -1 K -1 for FL MoS [19]. Compared to the extensive studies on the thermal conductivity of grapheme [20-23], comprehensive analysis of the Grรผneissen parameter, phonon relaxation time and phonon mean free path (MFP) of MoS is currently still lacking, despite their critical role in the understanding of the phonon scat-tering, temperature effect, and phonon-mode contribution to the intrinsic thermal conductivity. In this study, by using density functional perturbation theory (DFPT), we investigate the lifetime of phonons, and intrinsic ๐ of monolayer MoS by calculating the Grรผneissen parameters, fre-quency- and polarization-dependent phonon relaxation time and MFP. Positive Grรผneissen parameters for all the modes, in con-trast to the negative Grรผneissen of the low-frequency mode in graphene, are found for monolayer MoS . The calculated domi-nated MFP of MoS is around 18.1 nm. Based on non-equilibrium Greenโs function (NEGF) scheme, the intrinsic ๐ at room temper-ature is found to be around 23.2 Wm -1 K -1 . Our study shows that, owning to the SโMoโS sandwich structure, the single-layer MoS is dramatically different from the one-atom-thick graphene with respect to the structural stability, thermal expansion, vibrating anharmonic behavior and thermal conductivity. II. COMPUTATIONAL METHOD
The calculations of the interatomic force constants (IFC) and phonon dispersion are performed using the Quantum-Espresso code [24], within the local density approximation (LDA) of Perdew-Wang. We use the norm-conserving pseudopotential with energy (charge density) cutoff up to 70 Ry (700 Ry). The first Brillouin zone is sampled with a
30 ร 30 ร 1
Monkhorst-Pack grid. The vacuum region thickness is greater than 16 ร . The struc-tures are relaxed until the forces exerted on the atoms are less than 0.01 eV/ร . The optimized equilibrium lattice constant of monolayer MoS is 3.09 ร , smaller than the measured value of 3.16 ร [25] as LDA normally underestimates the lattice constant. In the following DFPT calculation, a Monkhorst-Pack
10 ร 10 ร q-mesh is used to calculate the dynamical matrix at each q grid, which gives the IFC through inverse Fourier transform to real space. The thermal conductance is calculated based on NEGF ap-proach [26,27]. The ballistic thermal conductance of a junction connected to two leads at different equilibrium heat-bath tempera-tures is given by the Landauer formula, ๐(๐) = โซ โ๐๐[๐] ๐๐ ๐ต (๐,๐)๐๐ ๐๐2๐โ0 (1) where ๐ ๐ต (๐, ๐) = 1/(๐ โ๐/๐ ๐ต ๐ โ 1) is the Bose-Einstein distri-bution function for a phonon with a frequency at the reserviors, ๐[๐] is the transmission coefficient, โ is Planckโs constant, and T is the average temperature of the hot and cold baths. Within the framework of NEGF, the phonon-transmission function ๐[๐] is given by
๐[๐] = ๐๐[๐บ ๐ ฮ L ๐บ ๐ ฮ R ] , where ๐บ ๐ and ๐บ ๐ are respec-tively the retarded and advanced Greenโs functions of the central region connected with two leads defined as ๐บ ๐ = (๐บ ๐ ) + =[๐ โ ๐พ ๐ โ ฮฃ L๐ โ ฮฃ R๐ ] โ1 with ๐พ ๐ being the force constant matrix and L(R) denoting the left (right) leads, ฮ ๐ผ ( ๐ผ = L and R ) is the broadening function describing the ability of phonons to enter and leave the leads and given by ฮ ๐ผ = ๐(ฮฃ ๐ผ๐ โ ฮฃ ๐ผ๐ ) , where ฮฃ ๐ผ๐ and ฮฃ ๐ผ๐ are the self-energies of the leads accounting for coupling of the central part with the leads. Here the IFC is directly calculated by first-principles. III. RESULTS AND DISCUSSION A.
Phonon dispersion
The lattice dynamics of bulk and monolayer MoS have been studied both experimentally [11, 28-30] and theoretically [31,32]. To facilitate the discussion of the mode-dependent scattering behavior and comparison with experimental results, as inspired by the previous work [33], we summarize the characters of the pho-nons for both the bulk (2H phase) and monolayer MoS in Figure 1a with respect to the symmetry assignment, frequency, optical character, and eigenvectors. Since the primitive cell of 2H-MoS and monolayer MoS contains 6 and 3 atoms, there are a total of 18 and 9 phonon modes, respectively. A factor group analysis of the point group (D and D for 2H-MoS and monolayer MoS , respectively) shows that the long-wavelength optical phonon modes at the ฮ point (without the three translational acoustic modes) can be decomposed as ๐ค ๐๐๐ก๐๐๐๐ (2H โ MoS ) = A (IR) + E (IR) + A (R) +2E (R) + E (R) + 2B (IN) + B (IN) + E (IN) ; (2a) ๐ค ๐๐๐ก๐๐๐๐ (monolayer) = A (IR) + E โฒ (IR + R) + A (R) +E โฒโฒ (R) (2b) where all the Raman (R), infrared (IR), and inactive (IN) modes are assigned. Note that the A , A , B , B , A ,A modes are singly degenerate and the E , E , E , E , E โฒ , E โฒโฒ modes are doubly degenerate. The R and IR modes are mutually exclusive in 2H-MoS due to the pres-ence of inversion symmetry in the crystal. The two IR modes in 2H-MoS , A and E , evolve into the IR active A and E โฒ modes in the monolayer case, respectively, where the latter is also Raman active due to the lack of inversion center in monolayer and assigned as E in bulk or FL MoS [11]. Another out-of-plane A Raman mode in 2H-MoS , which is normally used to identify the layer number of FL MoS flakes, matches with the lar A mode in the monolayer case, where the top and bottom sul-fur layers vibrate out-of-phase with direction normal to the basal plane while the Mo layer remains stationary. Our calculated fre-quencies for these modes listed in Fig.1(a) are in good agreement with experimental measurements [1,2]. FIG. 1. (a) Comparison of the zone-center modes between 2H-MoS and monolayer MoS . The frequencies of the 2H phase are adopted from Ref. [32]. (b) Phonon dispersion and phonon DOS for monolayer MoS . The polarization of all the phonon branches is presented. The low-frequency modes B and E below 60 cm -1 (denoted as B and E ) in 2H-MoS , have no cousin modes in the mono-layer. However, the two modes evolve into a series of shear (S) and compression (C) modes in FL MoS spreading around 30 cm -1 , respectively [34-37]. These low-energy optical phonons are easily thermally activated at room temperature and believed to greatly affect the carriers mobility and thermal conductivity via electron-phonon scattering and phonon-phonon scattering as simi-lar in FL graphene [38,39]. These interlayer S and C modes in FL MoS show a strong anharmonic character and layer-dependent frequency shift, and have been observed experimentally only recently [37]. We also predict a large Grรผneissen parameter of the low frequency acoustic modes in monolayer as shown below. The phonon dispersion and density of states (DOS) of the mon-olayer MoS are shown in Figure 1b. There are three acoustic branches: transverse acoustic (TA), longitudinal acoustic (LA), and out-of-plane transverse acoustic (ZA) branches, which are separated by a gap of around 50 cm -1 below the non-polar trans-verse optical (TO) and longitudinal optical (LO) modes, labeled as TO and LO , respectively. The TO , LO , and ZO modes are three polar branches. The homopolar ZO branch shows a nondispersive behavior, accompanying with a breathing mode eigenvector (see ๐ด mode in Figure 1a). In polar semiconductor or insulators, each IR active mode (polar mode) displays the LO/TO splitting due to the coupling of the lattice to the polarization field created by the polar mode in the long-wavelength limit. For bulk 2H-MoS , the Born effective charges (BEC) of Mo and S are small [33] and the polarized fields associated with the two IR modes ( A and E ) are weak (only leading to a 2 cm -1 LO/TO splitting [28]) due to a small mode oscillator strength [40,41]. For monolayer MoS , the electronic screening is weaker than 3D case and the splitting will be even smaller and thus neglected here. Table 1. Phonon parameters of single-layer MoS at ฮ for three acoustic modes (labeled according to polarization) and four optical modes (labeled according to symmetry assignment of irreducible representation of D point group): frequency (ฯ), Grรผneissen parameter (ฮณ) determined by biaxial strain, and relaxation time (ฯ). Note that ฮณ and ฯ for the three acoustic modes (ZA, TA, and LA) are shown at the cutoff frequency of 0.05 TH Z . ZA TA LA E โฒโฒ E โฒ A A ฯ(cm -1 ) 0 0 0 298.2 402.7 423.9 490.5 289.2 a a a a ฮณ 159.71 58.63 25.31 0.42 0.54 0.20 0.44 0.54 b c ,0.21 b ,0.6 d b b ฯ(ps) 6.93 22.17 48.21 16.84 5.50 38.36 5.72 5.1 e e a Ref. [32]; b Ref. [46]; c,d ฮณ obtained from Ref. [47] and Ref. [10], respectively, by measuring the shift of the Raman peak through applying uniaxial strain; e Relaxation time (ฯ) measured for the bulk 2H-MoS phase of the two IR-active modes: E and A modes corresponding respectively to the E โฒ and A modes of the monolayer case, by analyzing the damping constant of the image dielectric spectrum in Ref. [56]. In Figure 2, we calculate the phonon group velocity ๐ ๐ =๐๐ ๐ /๐๐ for the n th branch along the ฮ-M high symmetrical line. The ๐ โ ๐ and
๐ โ ๐ relationships are plotted in the Figure 2a and b, respectively. For the TA and LA modes along the ฮ-M direction, the sound velocities at long wavelength limit are about 693.5 and 1108.8 m/s, respectively, much smaller than the graphene case of 3743 (TA) and 5953 (LA) m/s [23]. For LA and TA modes, the group velocities drop dramatically with increasing phonon frequency, while for ZA mode, its group velocity increas-es with the frequency, reaches maximum at the q point sitting at the middle of the ฮ-M line, then decreases and finally reaches to zero at the zone edge. B. Grรผneissen parameter
The Grรผneissen parameter ( ๐พ ), which provides information on the anharmonic interactions between the lattice waves and the degree of the phonon scattering, is calculated by dilating the lat-tice with ยฑ ๐พ of each phonon mode at q point with s polarization is given by [42,43]: ๐พ ๐ชs = โ ๐2๐ s (๐ช) ๐๐ s (๐ช)๐๐ (3) where a is the relaxed equilibrium lattice constant of 3.09 ร . In Figure 3, the values of ๐พ for the acoustic and optical modes of MoS are plotted to reflect the mode-dependent strength of anharmonicity. The values of ๐พ for all the acoustic and optical modes at ฮ point are compiled in Table 1. The values of ๐พ for the optical modes are 0.42 ( E โฒโฒ ), 0.54 ( E โฒ ), 0.20 ( A ), and 0.44 ( A ), respectively, con-sistent with the measured value of Raman active E โฒ of 0.6 in Ref. [10]. It is interesting to find that all the modes show a positive value of ฮณ , indicative of a normal behavior of softening frequen-cies with expanding the lattice host [44]. The positive ๐พ indicates a positive coefficient of thermal expansion of monolayer MoS even at low temperatures where only the acoustic modes are ex-cited. This is in contrast to the negative thermal expansion ob-served in graphene at low temperatures due to the negative ๐พ of the ZA mode [45]. The issue of thermal expansion is critical for two-dimensional devices as the different thermal expansion coef-ficients between samples and substrate can result in strain, which may affect the performance and reliability for electronic device applications. Our results of Grรผneissen parameter ๐พ show that monolayer MoS is a credible alternative to graphene in some applications when a positive thermal expansion monolayer mate-rial is preferred. The underlying reason for the difference in ther-mal-mechanical relationship between MoS and graphene may be due to the suppression of the bending mode in MoS owning to its sandwiched structure. While the ๐พ of optical modes were reported previously [10,46,47], the ๐พ for the acoustic modes is still un-known. The values of ๐พ for the three acoustic modes at ฮ point are 159.7, 58.6 and 25.3 for ZA, TA, and LA, respectively, which are overwhelmingly larger than those of the optical modes within the whole Brillouin zone. These results show that the anharmonicity is weaker for optical modes than acoustic modes. FIG. 2. Wavenumber (a) and frequency (b) dependent phonon group velocity of acoustic phonon modes along ฮ-M direction. The variation of the frequencies for modes at ฮ point with compressive and tensile strains is plotted in Figure 3c, where the frequency shift with strain ( ฮด ) is defined as โ๐(ฮด) = ๐(ฮด) โ๐(0) . The different slopes of the ฮด - โ๐ curves reflect the different stiffening or softening behavior of each phonon mode under strain. For A mode, the slope is the smallest among all the modes, consistent with the smallest ๐พ of ZO branch within the whole Brillouin zone (Figure 3b). Eigenvector of this mode shows that the S atoms vibrate in counter phase in direction normal to the plane (Figure 1) and the Mo plane keeps stationary. Thus the frequency is relatively insensitive to the in-plane strain, but quite sensitive to the disturbance normal to the plane like increasing layers [32], electronic doping or chemical doping above the planes. In contrast, the E โฒ mode involves the in-plane vibration, and thus is more sensitive to the in-plane strain. Note here that the FIG. 3. The Grรผneissen parameter of acoustic (a) and optical (b) modes of monolayer MoS , the inset in (a) is a close-up view. (c) Frequency shift for optical modes under biaxial strain in the sheet. trends for these two modes are reversed in the case of doping on the layer, where the A mode shows significant softening behav-ior whereas the E โฒ mode remains nearly constant [12]. Our study of the larger slope of E โฒ mode (E for bulk MoS ) than the A mode (A for bulk MoS ) with strain is consistent with previous measurements [46,47]. Assuming a linear expan-sion of the sample with increasing temperature, the frequency shift can also be detected from the evolution of the Raman spec-trum for samples at different temperatures. The positive ๐พ is con-sistent with the redshift of the Raman-active modes with tempera-ture observed in experiments [9,19,48-50]. C. Relaxation time and mean free path based on Umklapp process
The phonon relaxation time in real materials can be obtained by the combination of various scattering processes, such as phonon-phonon Umklapp scattering, boundary scattering and defects scat-tering. By improving sample quality, extrinsic scatters such as defects and grain boundary can be removed. In the present work, we focus on intrinsic phonon relaxation process in monolayer MoS sheet, thus boundary scattering is not considered. The in-trinsic phonon relaxation time associated with phonon-phonon Umklapp scattering was derived using an expression given by Klemens based on the time-dependent perturbation theory [51,52], but introducing separate lifetimes ( ๐ ๐ชs ) for different pho-nons branches: ๐ชs = 2๐พ ๐ชs2 ๐ ๐ต ๐๐๐ ๐ ๐ชs2 ๐ ๐ (4) where M is the atomic mass and ๐ ๐ is the Debye frequency, T is the temperature, k B is the Boltzmann constant, and v is the av-eraged sound velocity given by the relation = 1/๐ฃ ๐ฟ๐ด2 + 1/๐ฃ
๐๐ด2 around the zone-center. It should be noted that a cutoff frequency ( ๐ ๐ ) for the long-wavelength acoustic phonons should be adopted in this formula to avoid the divergence issue [53,54]. In the previous study on bulk graphite, Klemens et al. [53] chose the frequency of the ZOโฒ mode at ฮ as the ๐ ๐ (around 4 THz) based on the assumption of the onset of cross-plane coupling and heat transport at this frequency. However, for monolayer MoS , this assumption cannot be applied as there is no ZOโฒ mode in the monolayer case. By non-dimensionalizing the data based on Umklapp scattering rates of phonons and assuming a constant sound velocity within the whole Brillouin zone, Freedman et al . recently showed an universal MFP spectrum in crystalline semi-conductors and revealed that approximately 90% of thermal con- ductivity is contributed from phonons with MFPs in the range between 1 and 200 times of the MFP of acoustic phonons at the Brillouin zone edge [55], covering the phonon frequencies be-tween 0.5% and 100% of the frequencies of acoustic phonons at the Brillouin zone edge. Based on this scenario, ๐ ๐ of 1.2 cm -1 is selected here which is 0.5% of the frequency of LA phonon at M point (~240 cm -1 ). Figure 4a gives the frequency dependence of relaxation time of LA, TA and ZA phonons in monolayer MoS sheet. For these acoustic phonon branches, the relaxation time decreases sharply as the frequency of phonons increases. The calculated ๐(๐ ๐ ) for both acoustic and optical branches are compiled in Table 1. The values for the ZA, TA, and LA modes are found to be 6.93, 22.17, and 48.21 ps, respectively. The ZA mode shows the shortest re-laxation time as an indication of the strongest phonon scattering, being consistent with the largest Grรผneissen parameter. For the four optical modes, the relaxation times are 16.84 ps ( E โฒโฒ ), 5.50 ps ( E โฒ ), 38.36 ps ( A ), and 5.72 ps ( A ). By fitting the classical dielectric oscillators with the measured IR reflectivity, Sun et al. recently obtained the relaxation times for the IR modes E โฒ (5.1 ps) and A (2.1 ps) [56]. With consideration of the phonon-phonon Umklapp scattering, our predicted ๐ of the in-plane E โฒ mode is in good agreement with the measured value (5.1 ps), whereas for the A mode the ๐ is 2-fold larger than the exper-imental value. For real materials, the phonon relaxation time can be decomposed into contributions from electron-phonon, phonon-substrate and phonon-phonon Umklapp scattering. For the out-of-plane vibrating A mode, the contribution from electron-phonon and phonon-substrate coupling is more significant than that in the in-plane E โฒ mode due to the d z character in the frontier orbitals [12]. This may account for the calculated larger phonon relaxation time for the A mode with only considering the Umklapp scatter-ing. For the Raman active phonons, unfortunately, there is no meas-ured value of the lifetime for comparison. Our result shows that the nonpolar optical E โฒโฒ (homopolar A ) mode possesses a much longer ฯ (three and seven times, respectively) than those of the IR-active E โฒ and A polar modes. This indicates that the thermal scattering in the optical modes mainly occurs through the polar modes. A fundamental understanding of the phonon MFP is essential to identify the phonon ๐ of two-dimensional materials. Based on the mode-dependent ๐ and group velocity, we obtain the Umklapp scattering limited MFP of monolayer MoS , which is a key quan-tity for understanding the size-scaling characteristics of ๐ . The phonon MFP for mode at q point with s polarization is defined as ฮป ๐ชs = V ๐ชs ฯ ๐ชs . It is well known that the acoustic modes contribute to most of the ๐ due to their relatively larger velocities than those of the optical modes [23]. Therefore, we only analyze the acoustic modes phonon MFP. The frequency- and mode-dependent pho-non MFP dominated by the Umklapp process is shown in Figure 4b. Similar to relaxation time, for all the phonon branches, MFP decreases obviously with increasing frequency, then increases and decreases slightly. Over the entire frequency regime, the MFP of LA mode is larger than those of TA and ZA modes, due to the combined effects from the frequency-dependent relaxation time and group velocities. To define an effective or frequency inde-pendent MFP for each branch, we employed the MFP at truncated frequency ( ๐ ๐ ) as the dominated MFP: Beyond which all phonons experience non-ballistic transport. The largest dominated MFP among the zone-center acoustic modes is found to be the LA mode with a value of around 18.1 nm, which is much larger than that of the TA mode (5.0 nm). This finding is consistent with the value of 5.2 nm by the recent MD calculation performed by Liu et al. [15] Our calculation shows that the MFP of monolayer MoS sheet is much smaller than that of the graphene ( โผ
775 nm) [20]. Since the sizes of most reported MoS flakes are around 1ฮผm [57-60], the thermal conduction in these samples is highly likely to be in the diffusive regime according to the predicted dominated MFP. Our study suggests that there appears to be a significant reduction of thermal conduction when the characteristic length, such as the size of sample or the size of grain, is reduced down to tens of nanometers as the grain boundary or edge boundary scat-tering becomes greatly enhanced. D. Thermal conductivity calculated from NEGF
FIG. 4. (a) Frequency dependent relaxation time and (b) mean free path for the three acoustic modes (ZA, TA and LA) of mono-layer MoS . In this part, we calculate the thermal conductance and thermal conductivity of monolayer MoS by using NEGF approach. The MoS sheet is modeled by creating an orthogonal supercell con-taining 144 atoms with assigning the armchair direction as the thermal conducting direction. Periodic boundary condition is applied in both the heat current direction and in-plane transverse direction. As it is computationally highly demanding by DFPT calculation for this system, here we use the Vienna ab initio simu-lation package (VASP) package [61] with the Perdew-Burke-Ernzerhof functional (PBE) to obtain the Hessian matrix by shift-ing individual atoms in the supercell and determine the forces induced on all other atoms. Figure 5 shows the obtained phonon transmission spectrum. The difference in mode frequency be-tween Figure 1 and Figure 5 due to the different approaches is small. The calculated thermal conductance is 1.06 nW/K for sam-ple with width of 1.27 nm. Assumed the thickness of the MoS sheet of 0.65 nm, the scaled thermal conductance, defined as thermal conductance per unit area ๐/๐ , amounts to 1.28 nWK -1 nm -2 . The thermal conductivity ๐ of a finite sample is related to the thermal conductance ๐ by ๐ = ๐๐/๐ , where l is the length of the sample, and S is its cross section. Within the ballistic regime ( ๐ < ๐ , ฮป is phonon MFP), the ๐ is linearly correlated with the length since the conductance ๐ is length-independent [21]. In contrast, when the sample length is beyond mean free path ( ๐ >๐ ), the transport is diffusive and ๐ is less sensitive to the variation of the length. It was proposed [21] that from the NEGF calculated ballistic thermal conductance, one may estimate the size-independent thermal conductivity of a sample with the relation ๐ = ๐๐/๐ , where ฮป is the phonon mean free path. Using this ap-proach, the room temperature thermal conductivity of single-layer graphene calculated from NEGF is 3410 Wm -1 K -1 [21], which is comparable with the experimental measurements [62]. Since thermal conductivity in semiconductors results mainly from pho-nons with long MFP [55,63], to derive the ๐ of MoS , here we use the dominated MFP of zone-center LA mode ( ๐ = 18.1 nm ) obtained in the lattice vibrational modes section. The room tem-perature (300 K) thermal conductivity of monolayer MoS is found to be 23.2 Wm -1 K -1 . It is worth mentioning that if the pho-non MFP of 5.2 nm (by the MD calculation in Ref. [15]) is adopt-ed, the ๐ is about 6.66 Wm -1 K -1 , which is in the same order of magnitude with the MD results of 1.35 Wm -1 K -1 [15]. FIG. 5. Phonon transmission for monolayer MoS calculated by the NEGF method. Our predicted ๐ of 23.2 Wm -1 K -1 of MoS is much smaller than the ultra-high thermal conductivity (4800-5600 Wm -1 K -1 ) of graphene [62]. Since the scaled thermal conductance ๐/๐ value of 1.28 nWK -1 nm -2 for MoS is only slightly smaller than that of graphene (~4.1 nWK -1 nm -2 from Ref. [64]) with the same width, the 100-fold reduction of ๐ in MoS arises from the much shorter MFP in MoS compared with graphene. Experimentally measured values of ๐ of FL MoS are from 0.4-1.59 Wm -1 K -1 [17,18], to around 52 Wm -1 K -1 by Raman spectros-copy approach [19]. This large discrepancy may originate from different sample quality, measurement methods and accuracy. Thus numerical simulations will definitely facilitate the under-standing of heat conduction in monolayer MoS sheet. MD has the obvious advantage that it does not rely on any thermodynam-ic-limit assumption, and is thus applicable to model nanoscale systems with real geometries. However, the calculated ๐ may rely on the inter-atom empirical potentials used. The combination of first-principles calculation and NEGF method has been commonly used to describe the ballistic phonon transport. In the present work, by adopting the Umklapp phonon-phonon scattering limited MFP, our calculation provides an estimation of the intrinsic ther-mal conductivity. However, the intrinsic relaxation time of pho-non-phonon Umklapp scattering is calculated from an empirical equation based on the time-dependent perturbation theory [51, 52]. Thus phonon scattering rates obtained fully from ab initio density functional theory calculations [65] without any adjustable parameters is still indispensable to determine the dominated pho-non MFP. IV.CONCLUSION
In-depth understanding on the lattice vibrational modes and in-trinsic thermal conductivity of MoS is highly important to speed up its potential applications in nanoelectronics and thermoelectric energy devices. Based on the DFPT calculations, we have ob-tained the Grรผneissen parameters and the Umklapp scattering limited relaxation time of phonons in monolayer MoS . The cal-culated phonon lifetime of the IR active modes is in good agree-ment with experimental results. The dominated phonon mean free path of the monolayer MoS sheet is around 18.1 nm, about 30-fold smaller than that of graphene. By using the NEGF approach combined with the obtained mean free path, the intrinsic thermal conductivity of monolayer MoS at room temperature is found to be around 23.2 Wm -1 K -1 , around two orders smaller than that of graphene, due to the significantly shorter phonon mean free path compared with graphene. ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support from the Agency for Science, Technology and Research (A*STAR), Singapore and the use of computing resources at the A*STAR Computational Resource Centre, Singapore.
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