High-temperature mechanical and thermodynamic properties of silicon carbide polytypes

Accepted Manuscript High-temperature mechanical and thermodynamic properties of silicon carbide polytypes Wei-Wei Xu, Fangfang Xia, Lijie Chen, Meng Wu, Tieqiang Gang, Yongfang Huang PII:

S0925-8388(18)32800-7

DOI:

10.1016/j.jallcom.2018.07.299

Reference:

JALCOM 47015

To appear in:

Journal of Alloys and Compounds

Received Date: 17 March 2018 Revised Date:

22 July 2018

Accepted Date: 25 July 2018

Please cite this article as: W.-W. Xu, F. Xia, L. Chen, M. Wu, T. Gang, Y. Huang, High-temperature mechanical and thermodynamic properties of silicon carbide polytypes, Journal of Alloys and Compounds (2018), doi: 10.1016/j.jallcom.2018.07.299. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

High-temperature mechanical and thermodynamic properties of silicon carbide polytypes Wei-Wei Xu1,*, Fangfang Xia1, Lijie Chen1, Meng Wu2, Tieqiang Gang1, Yongfang Huang1 1

School of Aerospace Engineering, Xiamen University, Xiamen 361005, P. R. China

Fujian Provincial Key Laboratory of Semiconductors and Applications, Collaborative

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2

Innovation Center for Optoelectronic Semiconductors and Efficient Devices, Department of Physics, Xiamen University, Xiamen 361005, P. R. China

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*

Corresponding author: [email protected] (Wei-Wei Xu)

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Tel: +86-592-2184310; Fax: +86-592-2182221

Abstract

Silicon carbide is widely used as ultra high-temperature ceramics, semiconductors,

and

radiation

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and pressure sensors with promising potentials for high-temperature, high-endurance, hardened

applications.

Daunting

difficulties

in

experimental

investigations of thermophysical properties hinder the better understanding of

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high-temperature material behaviors of silicon carbide. We present a comprehensive

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study of temperature-dependent mechanical and thermodynamic properties of SiC polytypes by first-principles methods. The obvious anisotropy of linear expansion and elasticity is found for 3C-SiC, while it is not distinct for other non-cubic SiC polytypes. Results show that the temperature dependences of mechanical properties exhibit the softening behavior, in which small linear reduction (~4.4%) in Vickers hardness and shear modulus but large linear reduction (~7.0%) in Young’s modulus are detected. The heat-resistant properties of SiC polytypes are ranked as 3C-SiC < 4H-SiC < 6H-SiC < 1

ACCEPTED MANUSCRIPT 15R-SiC < 2H-SiC. The present predictions are in favorable accord with available measured data in the literature.

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Keywords:

Silicon carbide; Mechanical properties; Thermodynamic properties; High temperature;

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Density-functional theory

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ACCEPTED MANUSCRIPT 1. Introduction Silicon carbide (SiC) has drawn widespread attentions in different domains [1]. Considering its wide energy bandgap, high thermal conductivity, and low thermal

temperature

circuit

operation

combining

with

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expansion, SiC is known as the most mature of semiconductors well-suited for high high-power,

high-frequency

environments in aerospace applications (e.g., turbine engines and the more electronic

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aircraft initiative) [2]. The high elevated-temperature stability, high thermal shock

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resistance and superior oxidation resistance favor SiC for use as ultra-high-temperature ceramics with its application in requiring high endurance, such as automobile brakes, nozzles, and friction bearings [3]. Besides these outstanding mechanical properties, SiC has also a low chemical reactivity with good corrosion resistance and irradiation

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resistance, thus it exhibits promising potential application in harsh environments [4, 5], such as geothermal well, space exploration, and nuclear power instrumentation. The combination of all these electrical, mechanical and thermal properties makes SiC an

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ideal materials for high-temperature pressure sensor devices [6].

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The underlying rationale for these SiC applications is the desirable combination of mechanical and physical properties, especially the high-temperature properties. However, the high-temperature materials behavior of SiC are anticipated to be much complicated not only because of their strong anisotropy but also due to the concurrence of various crystal polytypes, e.g., cubic (3C), hexagonal (2H, 4H and 6H), and rhombohedral (15R) SiC [7]. On the other hand, numerous daunting difficulties in experiments hinder the estimation of high-temperature properties of SiC, such as 3

ACCEPTED MANUSCRIPT specimen purity, thermal stress effects during the fabrication process, and the strong anisotropy. For instance, anisotropic thermal expansion leads to induced stress causing dislocations and can even break the crystal [8]. Concerning the specimen purity,

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measurements of thermal expansion [9], heat capacity [10], and elastic compliances [11] at elevated temperatures have been reported earlier. However, mostly polycrystalline and/or polytype mixture of SiC materials are adopted in these

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experiments instead of pure single crystal samples. To the best of our knowledge, there

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have been few experimental studies thus far on the fundamental mechanical and physical properties of SiC at elevated temperatures that cover various polytypes, although SiC is of particular interest in the aforementioned high-temperature applications.

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Density functional theory (DFT), combined with the quasi-harmonic approach, has been proven to be a valid methodology to estimate the high-temperature mechanical properties of materials in a number of systems [12-14]. An early work by Tang and Yip

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[15] predicted the thermo-mechanical properties of 3C-SiC by molecular dynamics

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simulations. But studies on the high-temperature mechanical properties of other primary SiC polytypes, including hexagonal 2H, 4H and 6H, rhombohedral 15R, are still scarce. To sufficiently explore the mechanical and physical properties at elevated temperatures for the typical SiC polytypes (i.e., cubic 3C, hexagonal 2H, 4H and 6H, and rhombohedral 15R), the present work is conducted using first-principles calculations and the quasi-harmonic Debye model.

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ACCEPTED MANUSCRIPT 2. Theoretical methods and details 2.1. Crystal structural information Silicon carbide owns a layered crystal structure and crystallizes in numerous

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different polytypes. All the polytypes are made up of a tetrahedral bonding configuration, arbitrarily either SiC4 or CSi4 [16]. There are three possible arrangements of atoms in a layer of SiC structure, called as A, B and C positions. The polytype is

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characterized by the stacking sequence of Si-C double-atomic layer. Figure 1 illustrates

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the positions of Si-C double-atomic layer in {112̅0} crystal planes for 2H, 3C, 4H, 6H, and 15R polytypes of silicon carbide [17] as previously described [18, 19]. 3C shows the zincblende lattice with a sequence of three steps in the same direction (stacking sequence ABC|ABC…, Fig. 1a). A unit cell of 2H with wurtzite lattice is obtained by a

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sequence of one step to the right and one step to the left (stacking sequence AB|AB…, Fig. 1b). The 4H and 6H with hexagonal unit cell show the stacking sequence ABCB|ABCB… (Fig. 1c) and ABCACB|ABCACB… (Fig. 1d), respectively. The stacking of

15R

with

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sequence

a

rhombohedral

unit

cell

is

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ABCACBCABACABCB|ABCACBCABACABCB… as plotted in Fig. 1e. A comprehensive description to SiC crystallography can be found in [16, 18, 19].

2.2. Quasi-harmonic Debye approach to thermodynamics Under the quasi-harmonic approach, the Helmholtz free energy F of a configuration at temperature T and volume V is given by [20]: 5

ACCEPTED MANUSCRIPT F(V, T) = E0(V) + Fvib(V, T) + Fel(V, T),

(1)

where E0(V) is the static energy at 0 K and volume V; Fel(V, T) is the thermal electronic contribution and Fvib(V, T) is the vibrational contribution to Helmholtz free energy. Note

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that Helmholtz free energy equals to Gibbs free energy at zero external pressure. E0(V) is calculated directly and then fitted using a four-parameter Birch-Murnaghan (BM4) equation of state (EOS) [20]. Fel (V, T) is determined by Mermin statistics as follows: εF

(2)

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Fel (V ,T ) = ∫ n(ε ,V ) f ε dε − ∫ n(ε ,V )ε dε + TkB ∫ n(ε ,V )[ f ln f + (1 − f )ln(1 − f )] dε ,

where n(ε, V) is the electronic density of states, f is the Fermi distribution, and εF is the

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Fermi energy. Fvib(V, T) can be obtained from either the quasi-harmonic Debye or the phonon approach. The recent studies [21, 22] confirmed that the former is sufficient to describe well the vibrational contribution to free energy and saves tremendous

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computing resources compared with the latter. Therefore, Fvib(V, T) was calculated using the Debye model in this study, which is estimated via

{

}

Fvib (V ,T ) = 9 8 ⋅ kB ΘD + kBT 3ln 1 − exp ( −ΘD T )  − D ( ΘD T ) ,

(3)

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where ΘD is the Debye temperature and kB is the Boltzmann's constant. The Debye x

function, D(x), is given by D(x) = 3 / x 3 ∫ z 3 / [exp(z) − 1]dz . To obtain Fvib(V, T), ΘD is the

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0

key parameter, which can be determined in terms of Debye-Grüneisen approximation [23] and Debye-Wang model [24]. Note that in the Debye-Wang model ΘD is described without the Grüneisen constant γ but with an adjustable parameter λ commonly taken 0, ±1/2, and ±1 (see [24] for more details).

2.3. Elastic properties at finite temperatures

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ACCEPTED MANUSCRIPT In an effort to calculate the second order elastic stiffness constants Cij’s, an efficient stress-strain method [25] is employed. The stresses and strains satisfy the Hooke’s law under small deformations and are related by

σ i = ∑ C ij ⋅ ε j .

(4)

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j

For a given set of strains, ε = (ε1, ε2, ε3, ε4, ε5, ε6) where ε1, ε2, ε3 refer to normal strains and ε4, ε5, ε6 refer to shear strains, are imposed on a crystal to generate the small

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deformations, corresponding one set of stress σ = (σ1, σ2, σ3, σ4, σ5, σ6) can be

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determined on the deformed lattice from first principles. Then the 6 × 6 elastic stiffness matrix C links E and Δ by C = E-1Δ where E and Δ are n × 6 strain and stress matrixes, respectively. To obtain Cij values, the following linearly independent sets of strains are applied [25],

0 0 0 x

0 0 x 0

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x    E =    

(5)

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x

0 0 0 0  0 0  , 0 0 x 0  x 

where each row is one set of strains with x being a normal (or shear) strain εi (which is

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small enough in the elastic range). To guarantee a reliable result, x = ±0.007 and ±0.013 are chosen, respectively, and the average values of elastic constants based on different x values are determined as suggested by the previous study [26]. Due to the crystal symmetry, there are three independent elastic constants (i.e., C11, C12 and C44) for the cubic structure and five independent elastic constants (i.e., C11, C12 C13, C33, and C44) for the hexagonal structure. To obtain high-temperature elastic constants, the quasistatic approach is adopted 7

ACCEPTED MANUSCRIPT [27]. This approach is based on the fact that the change of elastic properties is mainly caused by volume expansion at finite temperatures [28-30]. The procedure is summarized in three steps: (i) calculating the static elastic constants at 0 K as a function

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of volume Cij(V) using the stress-strain method [31]; (ii) predicting the volume change as a function of temperature at ambient pressure V(T, 0) using the Debye model [32]; (iii) then the temperature dependence of isothermal elastic constants can be derived,

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as Cij(T) = Cij(T(V)).

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The aggregate properties of bulk (B), shear (G) and Young's (E) modulus, and Poisson's ratio (ν) associated with polycrystals are estimated by means of the Voigt-Reuss-Hill (VRH) approximation [33, 34].

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2.4. First-principles calculations

All DFT-based first-principles calculations were performed by the projector augmented wave (PAW) method [35], as implemented in the Vienna Ab initio

the

generalized

gradient

approximation

(GGA)

parameterized

by

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within

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Simulation Package (VASP) [36, 37]. The exchange-correlation functional was treated

Perdew-Burke-Ernzerhof (PBE) [38, 39]. The standard valence shells and electronic configurations were used for Si and C. Wave functions were expanded in plane wave up to a cutoff energy of 500 eV for all the polytypes. Brillouin zone integrations were approximated by using a special k-point sampling of Monkhorst-Pack scheme [40] with Γ-centered 13×13×11, 15×15×15, 13×13×5, 13×13×3, and 7×7×7 grids for 2H, 3C, 4H, 6H, and 15R polytypes, respectively. To get a reliable result of elastic properties, a 8

ACCEPTED MANUSCRIPT higher dense k-mesh was set (e.g., 17×17×17 for 3C-SiC). Throughout the calculations, the convergence thresholds of total energy and the maximum force acting on ions

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were set to 10−5 eV/atom and 10−3 eV/Å, respectively.

3. Results and discussion 3.1. Bulk properties

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Table 1 summarizes the calculated 0-K equilibrium properties of the polytypes,

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including the lattice parameters (a0, c0), density (ρ), bulk modulus (BEOS) and its derivative of pressure (B’) fitted by the BM4 EOS, and the cohesive and formation energies (ΔE and ΔH), together with the available experimental data [41-47]. The calculated lattice parameters of polytypes agree very well with the measurements. The

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values reveal an average deviation less than 1.0%. The slight overestimation of lattice parameters results in the underestimation of density. The cohesive energy ΔE and formation energy ΔH are respectively defined as follows: (6)

∆ H ( SiC ) = [E t ( SiC ) − E s (Si ) − E s (C )] 2 ,

(7)

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∆ E (SiC ) = [E t (SiC ) − E a (Si ) − E a (C )] 2 ,

where Et(SiC) is the total energy (per unit cell) of SiC polytypes; Ea(Si/C) and Es(Si/C) are the total energy of corresponding elements as the isolated atom and in their most stable state, respectively. Herein, the diamond cubic structure for silicon (Si) and the layered planar structure for graphite (C) were chose as their most stable states. It is noteworthy that the total energy of graphite was computed with van der Waals corrections [48] due to the existence of dominated van der Waals forces between 9

ACCEPTED MANUSCRIPT adjacent planes in graphite. The obtained cohesive energy of graphite is -8.101 eV/atom, which is consistent very well with the previous theoretical data (-7.989 eV/atom) [49] while lower than the experiments (-7.374 eV/atom) [50]. As seen from

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Table 1, the calculated ΔE(3C-SiC) is in reasonable agreement with the only experimental result [42] with an error of 2.4%. Although the SiC polytypes own certain of different crystal structures, their exhibit a close values of cohesive and formation

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energies. Based on the calculated ΔH values, it is indicated that 6H has the most stable

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structure. The thermodynamic stability of the polytypes can be arranged as: 2H < 15R < 3C < 4H < 6H.

As the most mature of semiconductors well-suited for high temperature operations, the indirect bandgap of silicon carbides is a critical parameter. To obtain

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accurate bandgaps, the HSE06 hybrid functional [51] was adopted to determine the electronic band structure of silicon carbides. The room-temperature bandgaps (Eg) are listed in Table 1, as well as the corresponding valence band (Ev) and conduction band

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energies (Ec). The 3C polytype shows the minimum indirect Eg of 2.26 eV while the 2H

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shows the maximum Eg of 3.21 eV. All the calculated indirect Eg values reproduce the measurements [52-54] very well, with a deviation less than 3.5%. It is indicated that the Eg values increase with the increase of hexagonality of polytypes, and the trend qualitatively agrees with the observation [55].

3.2. Linear thermal expansion In terms of the Debye-Grüneisen approximation and the Debye-Wang model, the 10

ACCEPTED MANUSCRIPT linear thermal expansion (LTE) coefficients at zero pressure (αl) were computed with considering the thermal electronic contributions (EI), which is determined by,

αl =

1  ∂V (T )    3V0  ∂T 

(8)

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where V0 is the equilibrium volume. The calculated results for SiC polytypes are plotted in Fig. 2 and Fig. 3 together with the available data from experiments [9, 56, 57] and

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theories [58]. Note that the parameter λ in the Debye-Wang model are taken as 0 and ±1. The Grüneisen constant γ in the Debye-Grüneisen approximation were adopted

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based on the Poisson’s ratio (ν) [59]. The calculated γ are 1.031, 1.023, 1.028, 1.030, and 1.029 for 3C, 2H, 4H, 6H, and 15R polytypes, respectively.

For 3C-SiC, the thermal expansion observed by Li and Bradt [56] are often quoted in the literatures. It is indicted from Fig. 2 that the calculated LTE coefficients of 3C-SiC

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from Debye-Wang model with λ = 0 are closer to the available experiment and theory [56, 58] than those from the Debye-Grüneisen approximation as well as the

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Debye-Wang model with λ = ±1. Therefore, in this study the Debye-Wang model with λ = 0 was adopted as the first choice for computing the afterwards temperature

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dependence properties of SiC polytypes. In the 300-400 K temperature range as seen from Fig. 2, however, the calculated values deviate from the measurement [56] but show an excellent agreement with the previous theoretical data from a complicated eleven parameter rigid-ion model [58]. Taking αl at room temperature as example, the experimental data from Li and Bradt [56] is relatively large, 3.5×10-6 K-1, compared to the present calculated value of 2.504×10-6 K-1 and the previous calculated value of 2.47×10-6 K-1 [58]. For convenience, the predicted αl values at room temperature of SiC 11

ACCEPTED MANUSCRIPT polytypes are summarized in Table 1. It is worth nothing that our predicted data is in very well consistent with the measured data [60]. Figure 3 illustrates the LTE coefficient for the non-cubic SiC polytypes, including 2H,

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4H, 6H, and 15R. These non-cubic polytypes show the anisotropy of LTE coefficients, since the expansion of their lattice parameters along a and c axis with increasing temperatures are different. The anisotropic LTE coefficients along a and c axis (i.e., αa

1  ∂c(T )  1  ∂a(T )  .   ; αc =  c0  ∂T  a 0  ∂T 

(9)

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αa =

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and αc) are obtained by,

As seen from Fig. 3a, the 2H polytype exhibits an obvious anisotropy of LTE coefficients, in which αc is always ~15.9% larger than αa. The predicted linear expansion curves along a and c axis for the 4H, 6H, and 15R are almost overlap with each other,

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suggesting their unapparent anisotropy of LTE coefficients as illustrated in Fig. 3b-d. Our conclusion identifies well with the recently experiment [57], in which it confirms

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no distinct anisotropy of LTE coefficients for the 6H SiC. Contrast to the 2H polytype, the αc values for the 4H and 15R are only ~1.3% smaller than their corresponding αa

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values. On the other hand, the predicted LTE coefficients of 4H and 6H are underestimated with an average deviation of 10.7% for αa and 2.5% for αc from the measurement by Li et al. [9] while a larger deviation of 17.4% for αa and 18.2% for αc from the measurement by Stockmeier et al. [57], respectively, as shown in Fig.3b-c. This disparity may be ascribed to the neglect of anharmonic effects in the Debye approximation [13]. The specimen purity in experiments may be the another primary reason. For instance, as reported by Li et al. [9], the powder sample with unavoidable 12

ACCEPTED MANUSCRIPT impurities, such as Fe, Al, Ti, and V, are used to measure the thermal expansion of the 4H polytype. It is notable that the other factors may also lead to the exist of deviations, including the simplifying assumption of Debye model on the treating of the harmonic

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oscillation, the exchange-correlation functional approximation in the DFT calculation, as well as the polycrystalline specimen and surface oxidation in experiments.

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3.3. Thermodynamic properties

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The temperature dependence thermodynamic properties, such as heat capacity, and entropy, are the important database in thermodynamic modeling (e.g., CALPHAD method [61]) for developing the multicomponent phase diagram, as well as for understanding the thermal vibrational response of materials at elevated temperatures.

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Fig. 4 plots the heat capacity at constant pressure (Cp) and entropy (S) of the cubic 3C-SiC calculated by the Debye-Wang model and the Debye-Grüneisen approximation. The Cp and S is respectively estimated by [20]

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 ∂S   ∂F  2 2 S = −  + β BTV ,  ; C p = CV + β BTV = T  T ∂ ∂ T  V  V

(10)

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where Cv is the heat capacity at constant volume; β is the volume thermal expansion coefficient; B, T, and V are the bulk modulus, temperature and volume, respectively. The experimental data of Cp and S [62] are also shown in Fig. 4 for comparison. The predicted Cp and S values as a function of temperatures for the 3C-SiC from the Debye-Grüneisen approximation and the Debye-Wang model with λ = 0 are almost overlap with each other, and they reproduce the previous data [62] as shown in Fig. 4a-b. For the heat capacity, the temperature-dependent curve of Cv from the 13

ACCEPTED MANUSCRIPT Debye-Wang model with λ = 0 are also plotted in Fig. 4a. As it shows in Fig. 4a, Cv value is proportional to T3 at temperatures near 0 K and then tend to be a constant near 50 J/K/mol when the temperature above 1000 K, which satisfies the Dulong-Petit limit [63]

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(Cv = 3nR, n: the number of atoms per unit cell). The similar trend can be also observed from other aforementioned models, which is not illustrated for the reason of simplicity. Analogously, the T3 law exhibits at temperatures near 0 K for Cp, then the Cp

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monotonically increases with the increase of temperatures above ∼200 K. For the

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predicted entropy, there is minimal difference amongst the values from the Debye-Wang model and the Debye-Grüneisen approximation. Such difference becomes considerable large at high-temperature range, in which the maximum value of 11.4% is found at 1800 K.

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Figure 5 presents the heat capacity Cp for the non-cubic SiC polytypes, including 2H, 4H, 6H, and 15R. Similar to the case of thermal expansions as discussed above, only experimental data [10] is available for 4H and 6H SiC polytypes. It is indicated from Fig.

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5b that our predicted result from the Debye model is in excellent agreement with the

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measurement, especially at the low-temperature range of 300~600 K. For the 6H-SiC, the experimental data are scattered as shown in Fig. 5c due mainly to the purity of specimens. At low temperatures (< 500 K), the present predictions from the Debye models agree the experimental data from monocrystalline specimens, but slightly larger than that from the powdered specimens. With the increasing of temperatures, the predictions deviate from both the data from monocrystalline and powdered specimens. However, at the high-temperature range of 900~1800 K, our results 14

ACCEPTED MANUSCRIPT reproduce the measurement from the JANAF thermochemical tables (cross points in Fig. 5c) [10]. The variation trend of heat capacities as a function of temperatures for the non-cubic SiC polytypes is similar to that for the cubic 3C-SiC as aforementioned. It

3.4. Temperature-dependent mechanical properties

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and the results are provided in the Supplementary materials.

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is worth nothing that the entropies for the non-cubic SiC polytypes are also calculated

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Elasticity as a critical mechanical property provides implications for numerous material-related behaviors, such as interatomic bonding [14, 64], the criterion of mechanical stability and intrinsic ductility/brittleness [65, 66], and fracture toughness [67]. In terms of the strain-stress method, the 0-K elastic properties of SiC polytypes

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were calculated, including elastic constants (Cij), bulk moduli (B), shear moduli (G), Young’s moduli (E), Poission’s ratio (ν), and B/G value. Table 2 summarizes the calculated results, together with the available theoretical [68-73] and experimental

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[74-76] data for comparison. For the 3C-SiC polytype, our calculated elastic constants

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are ~6.1% in average smaller than the measured values [74]. Such underestimation of elastic constants can be also found in the 4H and 6H polytypes as listed in Table 2. Despite all this, the agreement between our predictions and the previous theoretical results is quite good. For instance, the deviation between our results and the other theoretical data from GGA-PBE potential [68] is less than 1% for the 3C-SiC. For the 2H-SiC, the previous theoretical data are scattered with a difference of ~13.2%. Our results are closer to the values from GGA-PBE [69] than those from LDA [71]. As 15

ACCEPTED MANUSCRIPT reported before, the LDA approximation exhibits the so-called “over-binding” effect [77] which always leads to the underestimation of lattice parameters and overestimation of elastic constants. Similarly, it is found for the 4H-SiC that our predictions are well

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consistent with the more recently theoretical data [69] but smaller than the measured values [75]. In a word, the present calculated elastic constants of the SiC polytypes are reliable, with an average error from the available data less than 10%.

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In terms of the universal anisotropy index (Au) [78], the elastic anisotropy is

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estimated. The Au is given by G V BV A = 5 R + R −6≥ 0, G B U

(11)

where G is the shear modulus, B is the bulk modulus, and the superscripts V and R represent the Voigt and the Reuss estimations of G and B, respectively. Au is zero for

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isotropic crystals, and the deviation of Au from zero defines the elastic anisotropy. One can see from Table 2 that the 3C-SiC exhibits the maximum elastic anisotropy with Au =

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0.49, followed by the 2H-SiC with Au = 0.17. The 6H and 15R polytypes own the minimum elastic anisotropy.

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By employing the elastic constants, the longitudinal (νl) and transverse (νm) elastic

wave velocities and the average sound velocity (νm) of SiC polytypes were calculated, and then the corresponding Debye temperature (ΘD) was obtained [14]. The calculated data are listed in Table 2. Comparing the ΘD values from the elastic wave velocity (ΘD(C)) and from the Debye-Wang model (ΘD(D)), it is found that the accordance is quite good with the maximum error of 0.8%. In addition, the predicted ΘD values also agree very well with the previous measured data [45, 60, 79, 80]. 16

ACCEPTED MANUSCRIPT Based on a theoretical model from the previous work [81], the Vickers hardness (Hv) of the SiC polytypes was estimated by Hv = 0.151G (G: shear modulus) due to the intrinsic brittleness of SiC. The predicted Hv for 3C-SiC is 28.2 GPa, which is reasonably

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consistent with the measured data (26±2 GPa [82]) but ~14.0% smaller than the theoretical (32.8 GPa [81]) value. It is indicated from Table 2 that the SiC polytypes exhibit almost the same Hv values around 28.0 GPa, in which 2H-SiC shows the

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minimum one (27.8 GPa).

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Thermoelastic properties are critical information for modeling thermal residual stresses and for optimizing the growth conditions of semiconductors thin films [72]. In terms of the quasistatic approach [27], the temperature dependence of isothermal elastic constants (CTij ) was calculated for the SiC polytypes from 0 K to 1800 K. Fig. 6

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illustrates the calculated results for the 3C-SiC, together with the experimental [56, 83] and theoretical [15, 69, 84] values for comparison. It is observed that the measured data are quite scattered and the previous theoretical results are scattered as well. The

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present results are closer to the experiment [56] than the previous results from

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molecular dynamics simulations [15]. The calculated CT12 values excellently agree with the measured results [56]. The present calculations underestimate the CT11 with an error of ~6.7% but overestimate the CT44 with an error of ~21.4%. Interestingly, although there are deflections in varying degree, the predicted curves of CTij yield almost the same slope with the experiment [56]. Fig. 7 plots the calculated temperature-dependent elastic constants for the non-cubic SiC polytypes. To the best of our knowledge, there are no measured temperature-dependent elastic constants C Tij but few elastic 17

ACCEPTED MANUSCRIPT compliances (STij = 1/CTij ) for these non-cubic polytypes. Therefore, the calculated elastic compliances STij are also plotted by dashed lines in Fig. 7. The measured elastic compliances as a function of temperatures [11] are included in Fig. 7 for comparison,

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as well as the available data of elastic constants [69-72, 75, 76]. For the 6H-SiC in Fig. 7c, it is found that our predicted CTij are in excellent agreement with the more recently theoretical results [72]. Although only few Cij at 298 K are observed [75, 76], our

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predicted ST11 and ST12 well reproduce the measured data [11] as shown in Fig. 7c, which

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validates the reliability of the present calculations. Similarly for the 4H-SiC (Fig. 7b), our predicted ST11 and ST12 values agree with the measured data [11] with a deviation less than 8.4%. In general, as illustrated in Fig. 7 the elastic constants Cij decreases linearly with the increase of temperatures, since the atomic interior bonding could be

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weakened by thermal expansion effects at elevated temperatures. The decreased amplitude (Δd) of CTij under the temperature range of 0~1800 K for the non-cubic SiC polytypes is tabulated in Table 3. For the Cij in different polytypes, the relative Δd(Cij)

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shows almost the same value. For instance, the C11 fell at an average rate of 10.8% in

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the non-cubic polytypes. It is observed that the reduction of the C13 is more pronounced than other Cij values in the non-cubic polytypes, while the C44 shows a smaller Δd than its counterparts. Based on these results, it can be drawn that the temperature factor may significantly change the mechanical deformation behavior of SiC polytypes due to its alteration of elastic constants in varying degrees. In order to further estimate the temperature dependent of mechanical properties, the isothermal elastic constants and elastic compliances at some selected 18

ACCEPTED MANUSCRIPT temperatures (i.e., 300, 900, 1500, and 1800 K) from Figs. 6-7 are list in Table 4. It is shown that all the properties (i.e., ΘD, Hv, E, and G) of the SiC polytypes decrease with the increase of temperatures. The decreased amplitude of ΘD for the 3C, 2H, 4H, 6H,

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and 15R polytypes are 2.2%, 1.7%, 2.1%, 1.9%, and 1.8%, respectively. The decreased amplitude of Hv for the 3C, 2H, 4H, 6H, and 15R polytypes are 5.0%, 3.9%, 4.7%, 4.3%, and 3.9%, respectively. The decreased amplitude of E (G) for the 3C, 2H, 4H, 6H, and

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15R polytypes are 7.0% (5.0%), 6.7% (4.1%), 7.3% (4.7%), 6.9% (4.5%), and 6.7% (4.2%),

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respectively. The 3C-SiC shows the maximum decreased amplitude of these aforementioned properties, implying its heat-resistant capacity is inferior to others non-cubic polytypes. Similarly, it is indicated that the mechanical properties of the 2H-SiC are much insensitive to temperatures than that of its counterparts. Based on

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these results, the conclusion can be drawn that the heat-resistant properties of SiC polytypes are ranked as: 3C-SiC < 4H-SiC < 6H-SiC < 15R-SiC < 2H-SiC. On the other hand, it is found that the high degree of elastic anisotropy for the 3C-SiC dramatically

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drops with the increase of temperatures. The decreased amplitude of Au for the 3C-SiC

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is 21.3%. For the non-cubic SiC polytypes, especially for the 2H-SiC, the elastic anisotropy seems to be insensitive to temperatures than that of the cubic 3C-SiC. The average decreased amplitude of Au for the 2H, 4H, 6H, and 15R polytypes are 13.2%, 16.4%, 16.2%, and 18.2%, respectively. To analyze the elastic anisotropy in more detail, the temperature-dependent Young’s modulus as a function of crystallographic orientation of SiC polytypes is calculated by employing the previous method [65]. The results are plotted in polar 19

ACCEPTED MANUSCRIPT coordinates as projected onto the (1̅10) crystal planes, as shown in Fig. S2 (see Supplementary materials). It is clear indicated that the 3C-SiC owns a butterfly-shaped curves with the maximum E value (Emax) along [111] crystal directions and the

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minimum E value (Emin) along [001] crystal directions. The non-cubic polytypes exhibit a star-shaped curves with the Emax along [001] crystal directions and the Emin along [111] crystal directions. Table 4 lists the Emax and Emin for SiC polytypes. For the cubic 3C-SiC,

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the Emax decreases sharply but the Emin reduces slightly with the increase of

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temperatures to narrow the different of E along [001] and [111] crystal directions, resulting in the significant decrease of elastic anisotropy. For the non-cubic SiC polytypes, such decreased degree of Emax and Emin with the increase of temperatures is fairly the same, leading to the insensitivity of temperature-dependent elastic

4. Conclusions

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anisotropy.

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For computational and data-driven development of silicon carbide materials, we

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have performed a study of the high-temperature mechanical and thermodynamic properties of SiC polytypes using first-principles calculations combined with the quasi-harmonic Debye model. A reasonable agreement is indicated amongst the predicted results, the experimental and theoretical data, which validates the reliability of the present calculations. The 2H-SiC exhibits an obvious anisotropy of linear expansion while no distinct anisotropy of linear expansions can be found for others non-cubic SiC polytypes. The 3C-SiC shows the maximum elastic anisotropy, followed 20

ACCEPTED MANUSCRIPT by the 2H-SiC, and the 6H- and 15R-SiC show the minimum elastic anisotropy. Such elastic anisotropy would be reduced to some extent with the increase of temperature. It is drawn that the heat-resistant capacity of the SiC polytypes can be ranked as: 3C-SiC

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< 4H-SiC < 6H-SiC < 15R-SiC < 2H-SiC, in which 3C-SiC is found to be inferior to those of other non-cubic SiC polytypes.

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Acknowledgements

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This work was financially supported by National Natural Science Foundation of China (Grant Nos. 51601161, and 51475396) and the Fundamental Research Funds for the Central Universities (Grant No. 20720170048). M. W. acknowledges the support by National Natural Science foundation of China (Grant No. 11704317) and China

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Postdoctoral Science Foundation (Grant No. 2016M602064). Y.F. H. acknowledges support from Natural Science Foundation for Young Scholars of Fujian Province (Grant No. JZ160402) and Scientific project for young and middle-aged teachers of Fujian

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province (Grant No. JAT160015).

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ACCEPTED MANUSCRIPT Figure Captions Fig. 1 (Color online) Schematic position of Si-C double-atomic layer in {112̅0} crystal planes for: (a) 3C, (b) 2H, (c) 4H, (d) 6H, and (e) 15R polytypes of silicon carbide [17].

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The large and small spheres are the silicon and carbon atom, respectively. Note that the colored spheres represent the periodical stacking sequences of polytypes (for example in 3C SiC, the stacking sequences is ABC|ABC...).

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Fig. 2 (Color online) Linear thermal expansion (LTE) coefficients (α) for the cubic 3C

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polytype of silicon carbide as a function of temperatures calculated by both of the Debye-Wang model (λ = ±1, 0) and the Debye-Grüneisen approximation plus thermal electronic contribution (EI). The available experimental data (open circles) from Li [56] and theoretical data (cross points) from Talwar [58] are also included for comparison.

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Fig. 3 (Color online) Linear thermal expansion (LTE) coefficients (α) for the non-cubic polytypes of silicon carbide: (a) 2H, (b) 4H, (c) 6H, and (d) 15R as a function of temperatures calculated by both of the Debye-Wang model (λ = ±1, 0) and the

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Debye-Grüneisen approximation plus thermal electronic contribution (EI). The available

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experimental data from Li [9] and from Stockmeier [57] are also included for comparison.

Fig. 4 (Color online) Heat capacity (Cp) and entropy (S) for the cubic 3C polytype of silicon carbide as a function of temperatures calculated by both of the Debye-Wang model (λ = ±1, 0) and the Debye-Grüneisen approximation plus thermal electronic contribution (EI). The experimental data from Barin [62] are also included for comparison. 25

ACCEPTED MANUSCRIPT Fig. 5 (Color online) Heat capacity (Cp) for the non-cubic polytypes of silicon carbide: (a) 2H, (b) 4H, (c) 6H, and (d) 15R as a function of temperatures calculated by both of the Debye-Wang model (λ = ±1, 0) and the Debye-Grüneisen approximation plus thermal

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electronic contribution (EI). The available experimental data from Hitova [10] are also included for comparison.

Fig. 6 (Color online) Calculated elastic constants (Cij) for the cubic 3C polytype of silicon

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carbide as a function of temperatures calculated by the Debye-Wang model (λ = 0). The

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available experimental data from Li [56] and Djemia [83] and theoretical results from Pizzagalli [69], Tang [15] and Varshney [84] are also included for comparison. Fig. 7 (Color online) Calculated elastic constants (Cij) and compliances (Sij) for the non-cubic polytypes of silicon carbide: (a) 2H, (b) 4H, (c) 6H, and (d) 15R as a function

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of temperatures calculated by the Debye-Wang model (λ = 0). The available experimental data from Kamitani [75], Arlt [76] and Karmann [11] and theoretical results from Sarasamak [71], Pizzagalli [69], Luga [70] and Reeber [72] are also included

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for comparison.

26

ACCEPTED MANUSCRIPT Table Captions Table 1 Calculated lattice parameters (a, c) (Å), density ρ (g/cm3), cohesive energies ΔE (eV/atom), formation energies ΔH (kJ/mol-atom), bulk modulus BEOS (GPa) and its first

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derivates with respect to pressure B’ (1/GPa), valence band energies Ev (eV), conduction band energy Ec (eV), bandgap energy Eg (eV), and linear thermal expansion coefficients (α̅l, αa, and αc) (10-6/K) for the polytypes of silicon carbide, together with

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the available experimental results [9, 41-47, 52-54, 60].

Table 2 Calculated elastic constants Cij (GPa), bulk modulus B (GPa), Young’s modulus E

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(GPa), the ratio of bulk modulus B and shear modulus G (B/G), Poisson ratio γ, elastic anisotropy index AU, longitudinal (υl), transverse (υt) and average (υm) sound wave velocities (m/s), Debye temperatures from elastic wave velocity [ΘD(C)] and Debye models [ΘD(D)] (K), and the Vickers hardness Hv (GPa) for the polytypes of silicon

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carbide. The available experimental [45, 60, 74-76, 79, 80, 82] and theoretical [68-73, 81] results are listed as well.

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Table 3 Decreased amplitude (Δd) for the elastic constants Cij of the non-cubic polytypes of silicon carbides (i.e., 2H, 4H, 6H, and 15R) under the temperature range of

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0~1800 K.

Table 4 Calculated isothermal elastic constants Cij (GPa) and elastic compliances Sij (1/GPa), the polycrystallines aggregates (E and G) (GPa), Debye temperatures from elastic wave velocity ΘD(C) (K), Vickers hardness Hv (GPa), elastic anisotropy index AU, and the minimum (Emin) and the maximum (Emax) Young’s modulus in [hkl] crystal directions (GPa) for the polytypes of silicon carbide at some selected temperature T (i.e., 300, 900, 1500, 1800 K). 27

ACCEPTED MANUSCRIPT Table 3 Decreased amplitude (Δd) for the elastic constants Cij of the non-cubic polytypes of silicon carbides (i.e., 2H, 4H, 6H, and 15R) under the temperature range of 0~1800 K. Δd(C13) 52.1 50.4 46.0 45.6

Δd(C33) 12.2 12.5 11.7 11.6

Δd(C44) 3.7 4.3 4.1 3.9

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Δd(C12) 31.9 34.2 31.9 31.5

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2H 4H 6H 15R

Δd(C11) 10.1 11.7 10.8 10.6

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C44 234.2 228.4 220.5 216.6 150.0 148.3 146.5 145.4 156.5 154.6 152.3 151.0 159.2 157.4 155.1 153.8 158.1 156.3 154.1 152.9

S11 3.156 3.197 3.253 3.286 2.159 2.211 2.281 2.323 2.206 2.266 2.346 2.393 2.205 2.265 2.340 2.381 2.209 2.260 2.332 2.374

S12 -0.750 -0.723 -0.687 -0.667 -0.415 -0.395 -0.370 -0.358 -0.431 -0.413 -0.387 -0.372 -0.436 -0.421 -0.398 -0.383 -0.438 -0.418 -0.395 -0.382

S13 -----0.147 -0.132 -0.112 -0.100 -0.154 -0.140 -0.121 -0.109 -0.156 -0.144 -0.126 -0.116 -0.157 -0.144 -0.127 -0.117

S33 ----1.951 2.014 2.101 2.152 1.958 2.023 2.110 2.162 1.946 2.007 2.090 2.137 1.948 2.010 2.092 2.139

S44 4.269 4.379 4.535 4.618 6.667 6.742 6.826 6.876 6.389 6.468 6.566 6.623 6.282 6.354 6.448 6.502 6.325 6.396 6.490 6.541

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C33 ----519.2 501.3 478.8 466.9 517.8 499.5 477.2 465.2 521.4 503.8 482.3 470.9 520.9 503.3 481.9 470.7

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C13 ----43.8 36.4 28.0 23.7 44.8 37.6 29.4 25.2 46.1 39.3 31.4 27.4 46.2 39.4 31.6 27.7

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C12 116.0 105.4 92.8 86.3 96.1 86.1 74.7 69.1 95.9 86.0 74.1 68.0 97.5 88.0 76.9 70.9 97.5 87.9 76.9 71.2

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C11 372.0 360.5 346.6 339.4 484.6 469.8 451.9 442.2 475.2 459.3 439.9 429.6 476.0 460.4 442.2 432.8 475.4 461.4 443.6 434.1

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Temp. 3C 300 900 1500 1800 2H 300 900 1500 1800 4H 300 900 1500 1800 6H 300 900 1500 1800 15R 300 900 1500 1800

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Table 4 Calculated isothermal elastic constants Cij (GPa) and elastic compliances Sij (1/GPa), the polycrystallines aggregates (E and G) (GPa), Debye temperatures from elastic wave velocity ΘD(C) (K), Vickers hardness Hv (GPa), elastic anisotropy index AU, and the minimum (Emin) and the maximum (Emax) Young’s modulus in [hkl] crystal directions (GPa) for the polytypes of silicon carbide at some selected temperature T (i.e., 300, 900, 1500, 1800 K). E 422.8 411.9 397.9 390.5 423.5 413.8 401.7 394.9 424.3 413.7 400.5 393.2 427.2 416.9 404.3 397.6 425.9 416.5 404.1 397.4

G 183.8 180.8 176.7 174.5 182.9 180.4 177.2 175.3 183.8 180.9 177.2 175.2 185.1 182.2 178.6 176.8 184.4 181.9 178.4 176.6

ΘD (C) 1138.8 1130.6 1119.9 1113.7 1136.2 1129.6 1121.3 1116.4 1145.2 1137.0 1126.5 1120.8 1141.9 1134.8 1124.9 1120.1 1140.0 1133.4 1124.4 1119.5

Hv 27.8 27.3 26.7 26.4 27.6 27.2 26.8 26.5 27.8 27.3 26.8 26.5 27.9 27.5 27.0 26.7 27.8 27.5 26.9 26.7

AU 0.45 0.42 0.38 0.36 0.16 0.16 0.15 0.14 0.12 0.12 0.11 0.10 0.11 0.11 0.10 0.09 0.12 0.11 0.10 0.10

Emin 316.8 312.8 307.4 304.3 380.9 373.1 363.6 358.3 389.2 380.7 370.0 364.0 393.8 385.5 375.2 369.6 392.0 384.2 374.1 368.7

Emax 506.4 489.4 467.7 456.6 512.6 496.5 475.9 464.7 510.7 494.4 473.9 462.6 513.9 498.2 478.5 467.9 513.5 497.6 478.0 467.6

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Table 1 Calculated lattice parameters (a, c) (Å), density ρ (g/cm3), cohesive energies ΔE (eV/atom), formation energies ΔH (kJ/mol-atom), bulk modulus BEOS (GPa) and its first derivates with respect to pressure B’ (1/GPa), valence band energies Ev (eV), conduction band energies Ec (eV), bandgap energies Eg (eV), and linear thermal expansion coefficients (α̅l, αa, and αc) (10-6/K) for the polytypes of silicon carbide, together with the available experimental results [9, 41-47, 52-54, 60].

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Source a0 c0 ρ ΔE ΔH BEOS B’ Ev‡ Ec‡ Eg ‡ α̅l αa αc 3C Present 4.380 3.170 -6.493 -13.807 212.2 3.9 7.32 9.58 2.26 2.504 a a b b c Expt. 4.358 3.22 -6.34 224 2.36 2.774d 2H Present 3.092 5.074 3.170 -6.490 -13.521 215.6 3.9 7.48 10.69 3.21 2.455 2.239 2.605 e e e f g Expt. 3.079 5.053 3.21 223.4 3.33 4H Present 3.094 10.129 3.172 -6.494 -13.862 215.6 3.9 7.40 10.56 3.16 2.466 2.455 2.439 h h h f g i Expt. 3.080 10.082 3.22 223.4 3.26 3.304 3.160i 6H Present 3.095 15.186 3.172 -6.494 -13.867 215.8 3.9 7.38 10.32 2.94 2.426 2.321 2.353 j j j f g i Expt. 3.081 15.125 3.21 223.4 3.02 3.356 3.246i 15R Present 3.095 38.236 3.150 -6.492 -13.778 216.0 3.8 7.39 10.28 2.89 2.419 2.319 2.308 k k k l Expt. 3.073 37.700 3.24 2.99 a b c d e 1965 Kawamura [41]; 1987 Chang and Cohen [42]; 2001 Goldberg et al [54]; 1975 Slack et al [60]; 1979 Schulz and Thiemann [43]; f1971 Yean and Riter [44]; g1996 Persson and Lindefelt [53]; h2009 Peng et al [45]; i1986 Li and Bradt [9]; j2007 Capitani et al [46]; k1944 Thibault [47]; l1981 Humphreys et al [52]. ‡ Note that the valence band energy Ev, conduction band energy Ec, bandgap energy Eg was obtained by the HSE06 hybrid functional method.

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Table 2 Calculated elastic constants Cij (GPa), bulk modulus B (GPa), Young’s modulus E (GPa), the ratio of bulk modulus B and shear modulus G (B/G), Poisson ratio γ, elastic anisotropy index AU, longitudinal (υl), transverse (υt) and average (υm) sound wave velocities (m/s), Debye temperatures from elastic wave velocity [ΘD(C)] and Debye models [ΘD(D)] (K), and the Vickers hardness Hv (GPa) for the polytypes of silicon carbide. The available experimental [45, 60, 74-76, 79, 80, 82] and theoretical [69-73, 81] results are listed as well.

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Refs. C11 C12 C13 C33 C44 B E B/G γ AU υt υl υm ΘD(C) ΘD(D) Hv 3C Present 383.4 126.8 240.4 212.3 433.4 1.14 0.16 0.49 7.67 12.06 8.44 1146.8 1146.8 28.2 a a a a Calc. 385.0 124.9 241.6 211.6 32.8k Calc. 212.5p 324.0p 1.12p 0.25p Expt. 390b 142b 256b 225b 448b 0.17b 1270f 26±2n 2H Present 492.1 101.5 49.5 531.8 151.0 213.0 428.9 1.16 0.16 0.17 7.62 12.02 8.38 1139.1 1146.8 27.8 c c c c c Calc. 499 93 52 533 153 h h h h 117 61 586 162h 229h Calc. 541 4H Present 486.6 103.4 50.7 531.7 157.9 212.7 431.8 1.14 0.16 0.13 7.65 12.05 8.41 1144.0 1152.4 28.1 Calc. 498c 91c 52c 535c 159c Calc. 534g 96g 50g 574g 171g 226g Expt. 501d 111d 52d 553d 163d 1194.8e 6H Present 485.0 104.1 50.8 533.0 160.4 212.7 433.2 1.14 0.16 0.12 7.67 12.06 8.43 1146.1 1155.3 28.2 i i i i i i Calc. 479.3 98.1 55.8 521.6 148.3 1123.8 j j j j 92 564 163 1200l Expt. 500 d d d d d Expt. 501 111 52 553 163 1080m 15R Present 485.6 103.9 50.9 532.5 159.2 212.7 432.4 1.14 0.16 0.12 7.69 12.10 8.45 1146.2 1155.8 28.1 a b c d e f 2013 Li et al [68]; 1991 Lambrecht et al [74]; 2014 Pizzagalli [69]; 1997Kamitani et al [75]; 2009 Peng et al [45]; 2002 Madelung et al [79]; g2007 Iuga et al [70]; h2010 Sarasamak et al [71]; i2014 Reeber et al [72]; j1965 Arlt and Schodder [76]; k2011 Chen et al [81]; l1964 Slack [80]; m1975 Slack et al [60]; n2001 Andrievski [82]; p2015 Zhao et al [73].

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Fig. 1 (Color online) Schemac posion of Si-C double-atomic layer in {1120} crystal planes for: (a) 3C, (b) 2H, (c) 4H, (d) 6H, and (e) 15R polytypes of silicon carbide [17]. The large and small spheres are the silicon and carbon atom, respecvely. Note that the colored spheres represent the periodical stacking sequences of polytypes (for example in 3C SiC, the stacking sequences is ABC|ABC...).

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Fig. 7 (Color online) Calculated elasc constants (Cij) and compliances (Sij) for the noncubic polytypes of silicon carbide: (a) 2H, (b) 4H, (c) 6H, and (d) 15R as a funcon of temperatures calculated by the Debye-Wang model (λ = 0). The available experimental data from Kamitani [75], Arlt [76] and Karmann [11] and theorecal results from Sarasamak [71], Pizzagalli [69], Luga [70] and Reeber [72] are also included for comparison.

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Highlights


The thermophysical properties of SiC polytypes were predicted by first principles. SiC polytypes exhibit the varying degrees of softening of mechanical properties.




The heat-resistant capacity of SiC polytypes are ranked as 3C < 4H < 6H < 15R <

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2H.

This study lays the foundation for high-temperature behaviors of silicon

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