Extremely Low Thermal Conductivity of Polycrystalline Silicene

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C: Physical Processes in Nanomaterials and Nanostructures

Extremely Low Thermal Conductivity of Polycrystalline Silicene Yufei Gao, Yanguang Zhou, Xiaoliang Zhang, and Ming Hu J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 05 Apr 2018 Downloaded from http://pubs.acs.org on April 5, 2018

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The Journal of Physical Chemistry

Extremely Low Thermal Conductivity of Polycrystalline Silicene Yufei Gao,1,2,† Yanguang Zhou,3,† Xiaoliang Zhang4,* and Ming Hu2,3,* ,# 1

School of Architecture & Civil Engineering, Shenyang University of Technology, Shenyang 110870, China 2 Institute of Mineral Engineering, Division of Materials Science and Engineering, Faculty of Georesources and Materials Engineering, RWTH Aachen University, 52064 Aachen, Germany 3 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, 52062 Aachen, Germany 4 Key Laboratory of Ocean Energy Utilization and Energy Conservation of Ministry of Education, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China

Abstract By performing Green-Kubo equilibrium molecular dynamics simulations, we study the thermal transport in polycrystalline silicene with grain size up to experimental scale of 50 nm and compare it with amorphous silicene. The thermal conductivity (TC) of polycrystalline silicene with small grain is not only lower than that of 1D polycrystalline silicon nanowires with the same grain size, but also lower than that of amorphous silicene. By introducing point defects, the TC of polycrystalline silicene with rather large grain size (30 nm) is comparable to polycrystalline silicon nanowires with extremely small grain size (2 nm). Through phonon spectral energy density analysis, we reveal that the ultralow TC of polycrystalline silicene †

Y.G. and Y.Z. contributed equally to this work. Author to whom all correspondence should be addressed. E-mail: [email protected] (M.H.) and [email protected] (X.Z.) # Present address: Department of Mechanical Engineering, University of South Carolina, Columbia, SC 29205, USA *

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originates from the violent phonon scattering exerted by the thin boundary of grains. For higher buckling distance, the phonon properties of amorphous silicene are prominently different from those of polycrystalline silicene and are closer to those of 3D bulk or 1D nanowire materials, and the scattering on the low-frequency phonons is weaker. In addition, the grain boundary mainly affects the long-wavelength phonons, while the point defects play dominant effect on highfrequency phonons. This study highlights the importance of 2D polycrystalline silicene for advanced thermoelectrics.


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Introduction Nowadays, the most famous method to fabricate high-quality two-dimensional (2D) materials is chemical vapor deposition (CVD). However, the as-formed 2D materials such as graphene and related materials produced by CVD1 inevitably exhibit polycrystalline morphology2,3, which can be seen as a patchwork of grains with different crystalline orientations. Undoubtedly, the grains will influence the relevant electrical and thermal transport in the structure4. For example, it will weaken the thermal transport properties of the original pristine structure5. However, the effect of grains is not always negative and it will strengthen the application of 2D structure in some fields. For example, an effective method to boost ZT, 6-8 which is the figure of merit used to characterize the energy conversion performance of thermoelectric materials, is to reduce the thermal conductivity (TC), and in the meanwhile maintain their electronic transport properties9-14. Therefore, the polycrystalline structure may show great potential in the realm of thermoelectricity than single crystalline structure. In the past decades, investigations on 2D polycrystalline structure were focused on graphene15-20 and little results concerned with other 2D structures21. However, elemental sheet of silicon (silicene) has emerged as a strong contender in the field of 2D materials22. Silicene23-28, which is the allotrope of silicon, shows distinct properties compared with graphene, and will have much more different applications. First, generally speaking, the conventional semiconductors have a significant band gap. However, graphene is zero-gap. This phenomenon unambiguously demonstrates that the missing gap will have important consequences for its application in electronic devices. Hence, in spite of graphene showing prominent electronic properties, it cannot be used as a conducting medium in conventional technology whose switching mechanism has a necessary relationship with the existence of an energy gap. In



contrast, silicene may have the ability to face this challenge for its electronic band structure and abnormal features such as spin-orbit coupling. In addition, it is well-known that silicene is a buckled structure which is different from planar graphene. This buckled structure will break the symmetry of out-of-plane direction29, and further give rise to a remarkable alteration of TC30-32. Therefore, silicene shows significant superiority to be an appealing candidate in the realm of thermoelectricity. Until now, although no investigations about the synthesis of polycrystalline silicene by experimental method were reported, an investigation33 has predicted the existence of polycrystalline silicene. In addition, considering that polycrystalline structures are common during the synthesis process of CVD, it is rather natural to expect that polycrystalline silicene can be produced in experiments in the future. In contrast, the existence of amorphous silicene has been proved by both experiment34 and density functional theory calculation35. Combining with the epitaxial growth of silicene on silver, the amorphous silicon film can be generated on the silver substrate when the temperature of substrate is lower than 450K34. Among different issues in polycrystalline models, the central one is how the TCs can be altered with the grain size. Previous molecular dynamics (MD) investigations were focused on individual grain boundaries36,37 or grains with small sizes (typically of a few nanometers38). However, considering the 2D polycrystalline structure produced by CVD usually contains grains with size of a few hundred nanometers and defects, accurate prediction of the thermal transport properties requires the consideration of relatively large grains and the effect of defects. In this paper, by performing Green-Kubo equilibrium molecular dynamics (GK-EMD) simulations and phonon spectral energy density analysis (SED), the thermal transport properties of polycrystalline silicene with grains whose size ranges from 5 to 50 nm, amorphous silicene and the effect of introducing single-vacancy point defects in related models are investigated. First,


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based on the TCs of polycrystalline silicene with different grain size, the formula that TCs scale with grain size is fitted, which can be used as an empirical equation in investigating the thermal transport properties of polycrystalline silicene. Meanwhile, it is interesting to observe that polycrystalline silicene shows extremely low TC. Its TC is not only lower than that of 1D polycrystalline silicon nanowires with the same grain size, but also lower than that of amorphous silicene. Moreover, the introduction of defects leads to a further reduction of TC: the TC of defective polycrystalline silicene with as large as 30 nm grains can drop down to the level of that for 1D polycrystalline silicon nanowire with extremely small grains (2 nm).

Model structures and computational methodology The structure of polycrystalline silicene is generated by Voronoi algorithm39. This is a top-down method. The construction details are listed as below: first, some seeds whose number equals to the grain number are randomly distributed in the plane. Then, the atomic configuration is generated around each seed with a random orientation. After that, a judgement about whether a generated atom belongs to the specific grain should be taken. The criterion is the distance between the generated atom and the seed in this grain should be the shortest one comparing with the distance with other seeds. If the criterion is not satisfied, the generated atom should be deleted. After the steps above, the polycrystalline silicene can be built as shown in Fig.1(a). In contrast, there are also some bottom-up methods, such as grain growth method15,17. In this method, polycrystalline structure is built from some small nucleation sites, some simulation methods such as MD are used to grow the polycrystalline structure from these initial blocks. As reported, there are many common experimental fabrication ways to produce monolayer nanostructure, such as epitaxial film growth40, thermal sublimation41, oxidation-reduction42 and 5 ACS Paragon Plus Environment


chemical vapor deposition (CVD)43. Corresponding to the monolayer nanomaterial produced by different experimental methods, the above two polycrystalline structure generating methods, i.e., Voronoi and grain growth methods, have their own superiority. In grain growth method, the under-coordinated atoms on the edges of grains can be handled much more meticulous, therefore, the stability of polycrystalline structure generated by grain growth method will be better, e.g., as reported in Ref.[15], the out-of-plane corrugation in the structures obtained by grain growth method is less obvious than that of Voronoi method. In addition, the shape and size of grain can also be controlled more easily in the procedure of grain growth method. Nowadays, as reported, both the shape and size of grain can be controlled well by the experimental conditions, and different shape of grain, e.g., flower-like, hexagonal and circular-shaped grains have been produced by experiments44,45. Therefore, it can be noticed that polycrystalline structures generated by grain growth method will be more coincided with these experimental nanocrystalline structures. As reported, this method has been successfully used for the generation of polycrystalline SiGe alloys46. In contrast, the polycrystalline structures generated by Voronoi method express much obvious randomicity, which are closer to the actual cases and more coincided with some uncontrolled experimental-produced polycrystalline structures. Next, based on the obtained initial configuration, the annealing procedure is taken to get the stable structure: first, the structure is relaxed at temperature of 600 K for 500 ps with NPT (constant pressure and temperature47,48) ensemble. Then, the structure is quenched from 600 K to 300 K within 1 ns under NPT ensemble. Next, another 500 ps is taken at 300 K with NVT (constant volume and temperature) ensemble. Finally, NVE (constant volume and energy) ensemble is used to further relax the structure. After these procedures, the final stable structure of polycrystalline silicene can be obtained and its atomic configuration can be seen in Fig.1(b).


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The structure of amorphous silicene49 is also built in this paper. It is obtained from the crystalline silicene by melting and annealing procedure. The size of crystalline silicene used to generate amorphous silicene is 50*50 nm2. During the construction procedure of amorphous silicene, first, the well-relaxed buckled silicene is heated from 50K to 3500K with a heating rate of 10 13 K ⋅ s − 1 and the whole system is run under NVT ensemble. In the meanwhile, the energy minimization is implemented every 20ps and it is done until the relative difference of energy is less than 10 − 12 . Then, when the system temperature reaches 3500K, the whole system is relaxed under NVT ensemble for 200ps to obtain the stable structure. During the next annealing procedure, the whole system is cooled down to 300K with a cooling rate of − 2 × 10 13 K ⋅ s − 1 , and the energy minimization process is implemented every 10ps. Next, the system is relaxed under NPT ensemble for 200ps to release the internal pressure. At last, another two 200ps are taken under NVT and NVE ensemble respectively, and the stable amorphous silicene can be obtained. In addition, during the whole procedure, a fixed boundary with elastic reflection behavior should be applied in the out-of-plane direction of the 2D structure, which is used to obtain the 2D amorphous structure. With the aim to judge the stability of amorphous silicene, the energies and temperatures of amorphous silicene under NVE ensemble are collected. Combining with these data, it can be noticed that the temperatures fluctuate in the range of 300 ± 3 K , and the energies are nearly kept at a constant value, the fluctuation range is less than 0.01eV. The above results can prove the amorphous silicene obtained in this paper is stable. In addition, we also find that the cooling rate in the annealing process shows significant effect on the structure of amorphous silicene. In this paper, combining with the rapid cooling ratio of − 2 × 10 13 K ⋅ s − 1 , the stable amorphous structure can be generated. In contrast, if the cooling rate is reduced to a relative low value of



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− 2 × 10 10 K ⋅ s −1 , the obtained structure is crystalline silicene and not amorphous structure,

which is coincided with the results in Ref.[49]. Stillinger-Weber (SW) potential50 is used to describe the interactions between silicon atoms in polycrystalline silicene. The specific parameters for silicene optimized by Zhang et al.51 are used. These parameters can accurately reproduce the buckled silicene structure and the phonon dispersion calculated from ab initio. The Large Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package52 is used to perform MD simulations. In addition, the timestep is set to be 0.5 fs throughout the entire simulations and the periodic boundary conditions are employed in the two lateral (in-plane) directions. The method used to calculate TCs is Green-Kubo equilibrium molecular dynamics simulation53. In this method, first, the heat flux is calculated with equation54

(

)

(

)

r r r r r 1 r r r r 1 J (t ) = ∑υiε i + ∑ rij Fij ⋅υi + ∑(rij + rik ) Fijk ⋅υi , 2 ij , i ≠ j 6 ijk i

(1)

where ε i , υr i and rrij present the energy and velocity of atom i and the distance between atom i

r r and j. Fij and Fijk present two- and three-body force, respectively. Then, after the heat flux gets stable, the autocorrelation calculation is taken on the heat flux to obtain the TCs of 2D nanomaterials

κ=

r ∞ r 1 J ( τ ) ⋅ J (0) dτ , 2VkBT 2 ∫0

where V , T, τ and k B

(2)

present volume, temperature, autocorrelation time and Boltzmann

constant, respectively. In the EMD calculation, the correlation time of our calculations is chosen to be 250ps and the timestep is set to be 0.5fs. The sizes of pristine silicene, amorphous silicene 8 ACS Paragon Plus Environment

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and polycrystalline silicene with grain size less than 20nm are 50*50nm2. When the grain size is larger than 20nm, the size of the polycrystalline model increases with the grain size. In this paper, the sizes of polycrystalline silicene with 20nm, 30nm and 50nm grains are 90*90nm2, 140*140nm2 and 200*200nm2, respectively. For each model of our simulation, 50 independent EMD calculations with different initial velocities are taken. During each calculation, heat current data within 3 ns are used to get TC. Finally, the results of 50 cases are averaged to obtain the final TC. Since classical MD is invalid for the structure whose Debye temperature is much higher than system temperature, quantum corrections should be taken on the TCs calculated by MD. The quantum effect is considered as the following equation51

TMD =

1 kB

∫

f max

0

 1 1 D( f ) + hfdf ,  exp(hf / k BTqc ) − 1 2 

(3)

where f, h and k B are phonon frequency, Planck constant and Boltzmann constant. T MD and Tqc are MD and quantum-corrected temperature, which are 300K and 230K in this paper, respectively. D ( f

) presents vibrational density of states (VDOS)55. Then, the TCs with

quantum effect can be written as

κ qc = κ MD

∂TMD , ∂Tqc

(4)

During the process of phonon spectral energy density analysis56-60, firstly, the phonon normal modes can be written as r Q& ( k ,ν , t ) =

∑ jl

r r mj r r r υ jl ( t ) ⋅ e *j ( k ,ν ) exp( − 2π ik ⋅ rl ) , N


(5)


r r

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r

* where e j ( k ,ν ) is eigenvector of eigenmode ( k ,ν ) . Then the spectral energy density is

calculated by taking Fourier transform

r

r

2

Φ (k ,ν , f ) = ∫ Q& (k ,ν , t ) exp(−2πift)dt ,

(6)

on the other hand, the expression of spectral energy density can be also written as another form58 r Φ ( k ,ν , f ) =

r C 0 ( k ,ν )

, r 2  2π [ f − f 0 ( k ,ν )]  r 1+   ( k ,ν ) Γ  

r

r

where C0 (k ,ν ) f 0 ( k ,ν ) and ,

(7)

r

Γ (k,ν ) present the mode-dependent constants, the frequency at

the peak center and the half-width at half-maximum, respectively. Combined with Eqs. (6) and

r

(7), Γ (k,ν ) can be obtained and the phonon lifetime can be calculated by fitting the function as

r

r

τ (k,ν ) = 1/ 2Γ (k ,ν ) . Finally, the TC of each phonon mode can be derived from the phonon Boltzmann transport equation (BTE)59,60

r

r

κα = ∑∑c phυ g2,α (k ,ν )τ (k ,ν ) , k

(8)

ν

where κ α , c ph and υ g ,α denote the TC in c ph = k B / V

α

direction, phonon specific heat calculated by

and phonon group velocity, respectively. By solving the dynamical matrices, phonon

dispersion curves can be obtained. And combining with equation υ g = ∂ω / ∂q , where ω and q present eigenfrequency and wave vector, respectively, the phonon group velocities can be calculated. In addition, it is generally recognized that the phonon distribution in classical molecular dynamics models is Boltzmann distribution, and in contrast, phonon obeys Bose10 ACS Paragon Plus Environment

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Einstein distribution in quantum correction models61. Considering the aim of SED calculation of this paper is to analyze the relative contribution of phonons with different frequencies which should not depend on the quantum correction coefficients, the application of quantum corrections will not affect the analysis conclusion obtained by SED, therefore, the quantum corrections for specific heat are not included in the SED procedure. In addition, participation ratio which is usually used to characterize the localized state of related phonon mode58 , can be calculated with equation Nb 3 r r r P( k ,ν ) = ( N b ∑ (∑ ei∗,α ( k ,ν )e i ,α (k ,ν )) 2 ) −1 ,

(9)

i =1 α =1

r

where

N b is the atom number and ei ,α (k,ν ) is the eigenvector defined above.

Results and discussion The data points in Fig.2 present the TCs of polycrystalline silicene with different grain sizes. It is found that the TCs of polycrystalline silicene are weakened with grain size decreasing. This is in good agreement with the conventional opinion that the decrease of grain size is equivalent to the increase of boundary, which will intensify the scattering between phonon and boundary, and further, gives rise to the reduction of TCs. On the other hand, more importantly, it can be also found that the TC of polycrystalline silicene with small grains can reach an extremely low value (about 1.36 W/mK, the reduced magnitude of TC taken by grain boundary is 82%). This value is not only nearly 100 times lower than that of polycrystalline graphene with the same grain size (nearly 150 W/mK)5, but also lower than the TC of polycrystalline silicon nanowires (SiNWs) with much smaller grain (2 nm), whose value and reduced magnitude of grain boundary are 1.44 11 ACS Paragon Plus Environment


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W/mK62 and 97.6%, respectively. More interestingly, it can be also noticed that the TCs of polycrystalline silicene with grain size smaller than 10 nm are even lower than the TC of amorphous silicene (about 2 W/mK, the dash dot line in Fig.2). Next, according to the phonon transport theory, the TCs, Kapitza conductance and grain size have the relationship as18,19

1 1 1 = + , κ (d ) κ 0 (d ) G ⋅ d where

κ

and

κ0

(10)

are the TCs of polycrystalline silicene and pristine silicene, G is Kapitza

conductance and d represents the grain size. Following Eq.(10), the specific TC vs. grain size curves calculated with different values of Kapitza conductance are shown in Fig.2. It can be observed that the data points obtained by GK-EMD show the same trend with the TC vs. grain size curves. Therefore, the credibility of the results of GK-EMD can be confirmed. Further, when the data points of GK-EMD in Fig.2 are best fitted to the formula of Eq. (10), we can notice that the value of Kapitza conductance of polycrystalline silicene is 0.3212 GW/m2K, which is nearly 100 times lower than that of polycrystalline graphene63. To test whether Kapitza conductance shows universality, two types of polycrystalline silicene with hexagon-shaped grain are built: one type is polycrystalline silicene with random-orientation grains and the other one is the orientations of grains set to be some specific angles. Combining with Eq.(10), the Kapitza conductance of the two type polycrystalline silicene can be fitted and the difference between them are nearly 20%. From these results, it can be observed that the distribution of orientation in polycrystalline silicene shows some effect in the thermal transport properties, which is similar to the conclusion in Ref. [64], and this effect needs further investigations. Therefore, it should be emphasized that the Kapitza conductance obtained from Fig.2 is the Kapitza conductance of


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polycrystalline silicene with random-shape and random-orientation grain. It is universal in some case, however, it is an approximation and can not present the Kapitza conductances of some specific models. Next, by using the fitted Kapitza conductance value, Eq. (10) can be transferred to the formula as κ (d ) =

2 .43 ⋅ d . 0 .32 ⋅ d + 7 .55

(11)

With Eq.(11), the TCs of polycrystalline silicene with different grain size can be obtained directly. Considering that the largest grain size of models in Fig.2 reaches 50 nm, which is close to the real grain size in experiments. Therefore, the Eq.(11) fitted by results in Fig.2 possesses extensiveness and can be used as an important empirical equation in investigating the thermal transport properties of polycrystalline silicene. Considering that the introduction of grains with too small size will weaken the stability of the model and result in the reduction of model’s application, and in addition, the existence of defect is ineluctable during the production process of silicene, little single-vacancy point defects are introduced in the polycrystalline silicene with 30 nm grain, aiming to realize the features of small grain model by one with large size grain. Fig.3 shows the relationship of TCs of polycrystalline silicene with 30 nm grain and defect ratio. The TCs of model decrease with the increase of defect ratio. When the value of defect ratio is 1.6%, the TC is weakened by nearly three times and drops down to 1.8 W/mK, which reaches the TCs of polycrystalline silicene with 5 nm grain (1.36 W/mK) and polycrystalline silicon nanowires (SiNWs) with 2 nm grain (1.44 W/mK)62. According to the related theory of defects, the TCs of crystalline solid and defect ratio are in accordance with the relationship65,66



κ=

κ0 1 + Rf ⋅ x

where

κ

,

and

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(12)

κ0 are the TCs of polycrystalline silicene and pristine silicene, x presents defect

ratio and R f is the reduced factor of defect. If the results in Fig.3 are fitted to the formula of Eq.(12), we can see the MD results (blue data points) and the curve of empirical formula (red curve) fits very well, which can prove the accuracy of our MD results. To analyze the mechanism of the reduced TCs with the introduction of grains and defects, and why the TCs of polycrystalline silicene is lower than that of amorphous silicene, four mode level phonon properties (squared phonon group velocity, phonon participation ratio, phonon lifetime and phonon mean free paths (MFPs)) of pristine silicene, polycrystalline silicene, polycrystalline silicene with defects and amorphous silicene are calculated and compared in Fig.4. First, we compare the phonon properties of pristine silicene, polycrystalline silicene and defective polycrystalline silicene. From Fig.4(a), we note that the average group velocity shows a huge reduction with the introduction of grains. The introduction of single-vacancy point defects gives rise to a further reduction of average group velocity. However, the degree of weakening is very little. As for participation ratio [Fig.4(b)], it can be observed that the participation ratio shows the same trend as phonon group velocity: grains lead to an obvious reduction and defects give rise to a further little reduction. Furthermore, we should note that the participation ratio of low-frequency phonons (0-4THz) in polycrystalline silicene and defective polycrystalline silicene reaches zero. It shows obvious lower value than that of medium- and high-frequency phonons, which means the localization of phonons in this region is stronger. To explain this phenomenon, the phonon properties of two dimensional nanosheet material (2D NSs), one dimensional nanowire material (1D NWs) and three dimensional bulk material (3D bulk) should 14 ACS Paragon Plus Environment

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be compared comprehensively. As reported by Refs. [62,67], for 1D NWs or 3D bulk, lowfrequency phonons show high MFPs and that of medium- and high-frequency phonons are obviously lower. However, in Fig.4(d), it can be found that the MFPs of different frequency phonons in pristine silicene, polycrystalline silicene and defective polycrystalline silicene are nearly the same. Specifically, the average MFPs of different frequency phonons in pristine silicene is nearly 10 nm and those of the other two models are 2 nm. All of the three MFP values are obviously higher than the thickness of silicene which is 0.42 nm. This phenomenon means that there will be violent scattering between boundary of silicene and phonons, especially for long-wavelength phonons. Therefore, we can draw the conclusion that the average MFPs of lowfrequency phonons which usually show long wavelength is weakened by the violent scattering excited by the boundary of 2D NSs and further drops to the same magnitude order of mediumand high-frequency phonons. On the other side, the strong scattering between boundary and lowfrequency phonons will enhance the localization of this part phonons, and therefore the phenomenon in Fig.4(b) can be understood. In general, phonon group velocity and participation ratio in Figs.4(a) and (b) have the same trend: the introduction of grains leads to the obvious reduction of the two parameters and that of defects gives rise to a further little reduction. However, comparing the results in Figs.4(c) and (d), there is no surprise to find that there are obvious difference between the trends of phonon lifetime or MFPs among the three models [Figs.4(c) and (d)] and those of group velocity or participation ratio [Figs.4(a) and (b)]. Specifically, the introduction of grains leads to the reduction of phonon lifetime and MFPs of all phonons, which is the same as that in Figs.4(a) and (b). However, the introduction of defects does not give rise to the alteration of phonon lifetime and MFPs of all phonons. Instead, the variation is focused on the high-frequency range, and in addition, the variation is much more



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obvious than that in Figs.4(a) and (b). This phenomenon can be explained as below: according to phonon theory, the phonon lifetime can be written as68

1 1 1 1 1 r = N r + U r + I r + B r , τ k,v τ k,v τ k,v τ k,v τ k,v

( )

where

τ,τ

( )

N

( )

( )

( )

(13)

, τ U , τ I and τ B present total phonon lifetime and relaxation time of N process, U

r process69, impurity70 and boundary scattering71, respectively. Then, k and ν are wave vector

and phonon branch. In Ref. [70], Klemens expresses the relaxation time of phonon-defect scattering in the form of

1 r = Aω 4 , τ k, v I

( )

(14)

where A is a constant related to the structure and its form can be seen in Refs. [72,73].

ω

presents angular frequency. According to Eq.(14), it can be observed that the reciprocal of defect scattering relaxation time is proportional to the fourth power of angular frequency. Therefore, we can understand the defect scattering will show much stronger effect on the relaxation time of high-frequency phonons. Further, as phonon theory points out, MFPs, group velocity and phonon

(r )

(r ) (r )

lifetime have the relationship as Λ k , v = υ g k , v τ k , v . Therefore, it can be also observed that the effect of defects on MFPs of high-frequency phonons will be more remarkable. Therefore, the phenomenon in Figs.4(c) and (d) that the phonon lifetime and MFPs of high-frequency phonons show more significant variation with the introduction of single-vacancy point defects can be understood. In general, according to the results in Fig.4, it can be summed up that the existence of boundary along thickness direction of 2D silicene gives rise to the huge alteration of properties of phonons with low-frequency and long-wavelength, and the introduction of grains


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leads to a further remarkable reduction of phonon properties of all phonons. In contrast, the introduction of defects only takes significant alteration on high-frequency phonons. Next, we compare the phonon properties of polycrystalline silicene and amorphous silicene. From Fig.4, it can be noticed that the four phonon properties of low-frequency phonons in amorphous silicene are obviously higher than those in polycrystalline silicene. In contrast, the corresponding properties of high-frequency phonons in amorphous silicene are lower in the same case. More importantly, the participation ratio of low-frequency phonons in amorphous silicene nearly reaches the level of medium- and high-frequency phonons, and the MFPs of lowfrequency phonons are remarkably longer than that of medium- and high-frequency phonons. In general, the variation trends of these two properties of amorphous silicene are prominently different from those of polycrystalline silicene and are closer to those of 3D bulk materials62. To prove these results, the well-relaxed structures of amorphous and polycrystalline silicene are compared. It can be found that the amorphous silicene shows higher buckling distance, which means its thickness is larger and its boundary scattering will be weaker. This phenomenon will lead to the higher phonon properties of low-frequency phonons in amorphous silicene comparing with polycrystalline silicene, which is in accordance with the results in Fig.4. In addition, it can be also noticed from the atomic configuration that the atoms in amorphous silicene usually distribute as multi-membered rings, which can be seen as a form of defect. Based on the analysis above, the effect of defect focuses on high-frequency phonons, therefore, the phenomenon that high-frequency phonons in amorphous silicene show weaker phonon properties (Fig.4) comparing with polycrystalline silicene can be understood. Finally, the results in Fig.4 can be used to analyze TCs of four models. Based on Eq.(8), we can find that the phonon group velocity, phonon lifetime and MFPs are all proportional to TCs.



In addition, participation ratio presents the localization of phonons. Usually, the higher the value of participation ratio, the stronger the delocalized status. Since delocalized modes are more effective than localized modes in facilitating heat transport74, the higher value of participation ratio also presents larger TCs. Then, as shown in Fig.4, the four phonon properties decrease with the introduction of grains and defects, and therefore, these results unambiguously demonstrate that the TCs will weaken in the same case, which is the same as the conclusion in Figs.2 and 3. Furthermore, comparing with 3D polycrystalline silicon and 1D polycrystalline SiNWs, the existence of thickness boundary in 2D polycrystalline silicene gives rise to severer boundary scattering and much more obvious reduction of properties of low-frequency phonons. Therefore, the polycrystalline silicene with large grain will have the same TC as polycrystalline SiNWs or polycrystalline silicon with small grain. At last, the four phonon properties of low-frequency phonons in amorphous silicene are obviously higher than those of polycrystalline silicene. This will lead to the higher TCs of amorphous silicene compared with polycrystalline silicene, which is also shown in Fig.2. With the aim to analyze the contribution of different phonons, the accumulated thermal conductivities as a function of frequency and MFP are calculated by SED and shown in Figs.5. The TC value of specified frequency (MFP) point presents the TC contributed by phonons whose frequencies (MFPs) are lower than this specified frequency (MFP). Therefore, according to Fig.5, the contributions of phonons with different frequencies (MFPs) to the total TC can be analyzed. It can be observed directly from Fig.5(a) that the TCs of pristine silicene, polycrystalline silicene and defective polycrystalline silicene are mainly contributed by medium- and high-frequency, the contribution of phonons from 4 to 18THz is nearly 90% of the total TCs. However, in contrast, the low-frequency phonons show a dominant contribution to the TCs of amorphous


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silicene, the contribution of phonons in the range of 0-1THz is more than 95% of the total TCs. These results are mainly due to the smaller buckling distance of the first three 2D models comparing with amorphous silicene. It will give rise to severer boundary scattering, which mainly concern the phonons with low-frequency and long-wavelength, and finally lead to the small contribution of low-frequency phonons in these three models. In addition, comparing the accumulated TC curves of polycrystalline silicene and defective polycrystalline silicene, we can notice that the difference mainly focuses on high-frequency range. This phenomenon can prove the previous conclusion that the defect shows more obvious effect on high-frequency phonons. Furthermore, to understand the mechanism, group velocity and phonon lifetime which produces direct effect on TCs according to Eq.(8) should be compared. In Figs.4(a) and (c), it can be observed that the difference of phonon group velocities between the two models distributes in the whole frequency range and that of phonon lifetime focuses on the high-frequency range. This result unambiguously demonstrates that the dominant effect of defects is the alteration of phonon lifetime, which means the main effect is caused by phonon scattering. Combining with Fig.5(b), first, we can notice that the TCs of pristine silicene, polycrystalline silicene and defective polycrystalline silicene are mainly contributed by shortand medium-wavelength phonons and the contribution of long-wavelength phonons is minor. For example, the contribution of phonons with wavelength from 20 to 30nm only occupies 3% of the total TCs of pristine silicene. This is obviously different from 3D bulk or 1D NWs model whose contribution of long-wavelength phonons is conspicuous62,67. Based on the analysis on the previous part, it can be understood that this difference is mainly due to the violent phononboundary scattering present in 2D silicene. Corresponding to the above three models, in amorphous silicene, the contribution of phonons with MFPs in the range of 200-300nm to the



TCs reaches to 20%, which is obvious higher than those of other three models and is nearly equal to the values of 3D bulk and 1D NWs. In addition, it can be also found that phonon MFPs range of pristine silicene is 0-30nm and that of polycrystalline silicene is 0-3nm. We can see the MFPs of polycrystalline silicene is nearly 10 times lower than that of pristine silicene in longwavelength range, which signifies the grain boundary shows more noticeable effect on longwavelength phonons.

Conclusions In summary, by performing Green-Kubo equilibrium molecular dynamics simulations, the TCs of polycrystalline silicene with comprehensive grain size (5 – 50 nm) and amorphous silicene are calculated. Our MD results demonstrate that the polycrystalline silicene shows extremely low TC: (1) Comparing with other polycrystalline silicon nanomaterials, polycrystalline silicene has lower TC in the same grain size range. This phenomenon is mainly caused by severe boundary scattering present in 2D polycrystalline silicene. Further, with the introduction of single-vacancy defects, the TC of polycrystalline silicene with substantially large grain size (30 nm) can drop down to the same level of TC of polycrystalline SiNWs with extremely small grain size (2 nm). (2) The TC of polycrystalline silicene with small grains is obviously lower than that of amorphous silicene, which is usually seen as the amorphous limit. By comparing the phonon properties, it can be observed that the higher buckling distance of amorphous silicene leads to the weaker boundary scattering with low-frequency phonons, and it finally gives rise to the higher TC of amorphous silicene. In addition, the TC of amorphous silicene is mainly contributed by phonons with low-frequency and long-wavelength, while the contribution of these phonons in polycrystalline silicene is very little. Finally, by performing SED, it can be also observed that the 20 ACS Paragon Plus Environment

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effect of grain boundary is mainly focused on long-wavelength phonons, while the effect of point defects is focused on high-frequency phonons, which can be seen as an effective approach to modulate the thermal transport properties of nanomaterials.



Acknowledgments Y.G. acknowledges the financial support from National Natural Science Foundation of China under Grant No. 11602149, Doctor Startup Foundation from Science and Technology Department of Liaoning Province under Grant No. 201501083, Department of Education of Liaoning Province under Grant No. L2014036 and Shenyang University of Technology under Grant No. 005636. Y.G. gratefully acknowledges the scholarship of the State Scholarship Fund of the China Scholarship Council under No. 201508210271 that supports him as a visiting scholar at RWTH Aachen University and contributes to this work. X.Z. acknowledges the support from National Natural Science Foundation of China (No. 51720105007) and the Fundamental Research Funds for the Central Universities (No. DUT16RC(3)116). Simulations were performed with computing resources granted by the Jülich Aachen Research Alliance-High Performance Computing (JARA-HPC) from RWTH Aachen University under Project No. jara0155 and jara0160 and computing resources from the Supercomputer Center of Dalian University of Technology.


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Fig.1 Structures of (a) initial (schematic) and (b) relaxed polycrystalline silicene with grain size of 30 nm.


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Fig.2 Thermal conductivities of polycrystalline silicene as a function of grain size (black data points). The solid lines from top to bottom are fitted by Eq.(10) with values of Kapitza conductance from 5.5 to 0.2 GW/m2K. The dash dot line presents the thermal conductivity of amorphous silicene.



Fig.3 The relationship between the thermal conductivities of defective polycrystalline silicene and defect ratio. The average grain size is 30 nm.


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Fig.4 Comparison of mode level phonon properties: (a) squared phonon group velocity, (b) participation ratio, (c) phonon lifetime and (d) phonon mean free path among pristine silicene, polycrystalline silicene, defective polycrystalline silicene and amorphous silicene.



Fig.5 Accumulated thermal conductivities as a function of (a) frequency and (b) phonon mean free path for pristine silicene, polycrystalline silicene, defective polycrystalline silicene and amorphous silicene.


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Table of Contents Graphic


Extremely Low Thermal Conductivity of Polycrystalline Silicene

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