Phonon optics, thermal expansion tensor, thermodynamic and chemical bonding properties of Al₄SiC₄ and Al₄Si₂C₅: a first-principles study

Abstract

The phonon spectra, chemical bonding, thermal and thermodynamic properties of Al₄SiC₄ and Al₄Si₂C₅ are calculated by first-principles using density functional theory. Raman and infrared (IR) active phonon modes and their eigenvectors are analyzed. Phonon mode-Grünseisen parameter and macroscopic Grünseisen constants are evaluated from phonon spectra. Employing quasiharmonic approximation (QHA), the thermal expansion tensor is obtained. The calculated volumetric thermal expansion coefficient (TEC) of Al₄SiC₄ is high than that of Al₄Si₂C₅; and the linear TEC in the [001] direction is slightly higher than that of the [100] direction for both compounds. The computed average linear TECs for Al₄SiC₄ and Al₄Si₂C₅ are 9.96 × 10⁻⁶ K⁻¹ and 8.9 × 10⁻⁶ K⁻¹ in a range from room temperature to 1500 K, respectively. Other thermal properties such as specific heats (C_V, and C_P), entropy (S), isothermal and isobaric bulk moduli (K_T and K_S) are also discussed. Using Slack's model, it is found that thermal conductivities are 59.9 W m⁻¹ K⁻¹ and 78.3 W m⁻¹ K⁻¹ at room temperature for Al₄SiC₄ and Al₄Si₂C₅, respectively. The response of chemical bonds to hydrostatic pressure is discussed using Milliken population analysis. We also plot the charge density and its reduced density gradient to reveal the covalent bonds in the structure.

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1. Introduction

Aluminum-containing ternary ceramics (Al₄SiC₄ and Al₄Si₂C₅) are regarded as low cost potential materials for high-temperature structural devices, heat-exchange materials, and weight sensitive applications.^1–5 The oxidation behavior of Al₄SiC₄ was investigated by Yamamoto et al.,⁴ who pointed out that the remarkable oxidation occurred in temperature range of 1123 K to 1423 K. Few years ago, thermodynamic behaviors of the same material were investigated through thermal conductivity^5,6 and electrical resistivity⁷ measurements. Wen and Huang⁵ provided the linear thermal expansion coefficient (TEC) of Al₄SiC₄ of 6.2 × 10⁻⁶ K⁻¹ in the temperature range from 473 K to 1723 K; they also gave the bending strength of 297.5 MPa and vickers hardness of 10.6 GPa. Inoue et al.⁶ claimed the thermal conductivity and linear TEC of highly densitied Al₄SiC₄ ceramic from room temperature to 1473 K were 80 W m⁻¹ K⁻¹ and 7.16 × 10⁻⁶ K⁻¹, respectively. Inoue and Yamaguchi⁷ studied the temperature dependence of electrical resistivity of bulk Al₄SiC₄ material, and they concluded that the electrical resistivity exhibited as an insulator. Al₄Si₂C₅ is another stable phase below 2310 K and mechanical properties of Al₄Si₂C₅ are comparable to these of the dense Al₄SiC₄ bulk phase.⁸ The crystallographic parameters of Al₄Si₂C₅ were provided in detail by Inoue et al.¹ Linnarsson et al.⁹ found that Al₄Si₂C₅ could form from a supersaturated solution of Al-doped 4H–SiC at 1700–2000 °C. Meanwhile, the chemical bonding, optical and elastic properties of Al₄SiC₄ and Al₄Si₂C₅ have been reported.^10,11 The band gaps of 1.05 eV and 1.02 eV were claimed for Al₄SiC₄ and Al₄Si₂C₅ by Hussain et al.¹⁰ Liao et al.¹¹ investigated the atomic displacements under deformation and the theoretical strengths of Al₄SiC₄, they suggested that structural failure occurs in tensile deformation firstly.

Although the ternary aluminum silicon carbides are of great importance. To our knowledge, there is less information available about the phonon optics, Grüneisen parameters, heat capacity, entropy, temperature dependence thermal expansion and thermal conductivity, especially for Al₄Si₂C₅. In present work, we will perform systematic studies on the phonon properties, thermodynamic parameters and chemical bonding behaviors.

2. Calculation methods and models

2.1. Density functional theory calculations

The ternary carbides discussed here have hexagonal structure (P6₃mc for Al₄SiC₄, R-3m for Al₄Si₂C₅),¹ and the conventional crystal structures (2 f.u. of Al₄SiC₄, and 3 f.u. of Al₄Si₂C₅) were employed here; the detailed structural information of which can be found in ref. 10 and 11 The all first-principles calculations were performed using the plane wave basis and Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional of generalized gradient approximation (GGA) as implemented in CASTEP code.^12–14 The Broyden–Fletcher–Goldfarb–Shannon (BFGS) algorithm was applied to relax both lattice parameters and atomic fractional coordinates. The kinetic energy cut-off value of 700 eV was used for plane wave expansion in reciprocal space. The ultra-soft pseudopotentials (USPPs) were employed to describe the Coulomb interactions between the valence electrons and pseudo-ion core. The valence electrons of USPPs were: Al: 3s²3p¹, Si: 3s²3p², C: 2s²2p², respectively. The sampling k-point mesh in the first irreducible Brillouin zone was generated by Monkhorst–Pack method,¹⁵ using a 6 × 6 × 1 grid for each structure. The convergence criterion for electronic self-consistency was 10⁻⁸ eV per atom for total energy, and the forces per atom were reduced to 10⁻⁴ eV Å⁻¹.

2.2. Thermodynamic properties and thermal expansion tensor

Furthermore, in order to calculate the thermal expansions, the Helmholtz free energy at the elevated temperature was calculated, which is given by F(V, T) = E_gs(V) + F_vib(V, T) + F_ele(V, T).¹⁶ The E_gs represents ground state total energy. The vibration free energy (F_vib) is calculated by phonon spectra in standard quasiharmonic approximation (QHA).

Within an appropriately normalized phonon density of states (PHDOS) . In eqn (1), N the total number of atoms in the unit cell, and ω is the angular frequency.

In this work, the phonon spectra were calculated in CASTEP code by finite element displacement method by setting a cutoff radius of 3 Å, which resulted in 2 × 2 × 1 supercell structure; and the LO–TO splitting was ignored. F_ele is the thermal electronic contribution to Helmholtz free energy, which is obtained by Mermin statistics F_ele = E_ele − TS_ele.

Minimizing the Helmholtz free energy F(V, T) with respect to cell volume V as a function of temperature gives the equilibrium volumes at different temperatures. This can be easily realized by fitting well known Birch–Murnaghan equation of state.¹⁷ Finally, the volumetric TEC λ(T) can be determined by , where V₀ and V₁ represent the equilibrium volumes at temperatures T₀ and T₁, respectively.

The Grüneisen parameters of a crystal structure indicate its anharmonic effects in the phonon spectrum due to the change of cell volume.¹⁸ In this paper, the phonon-mode Grüneisen constant (γ(ω_i)) and macroscopic Grüneisen parameter (γ) were calculated. The former one describing the phonon frequency shift with respect to volume can be calculated by:

where, ω_i represents the phonon mode i with angular frequency ω. The macroscopic Grüneisen parameter is defined as the weighted average of the mode Grüneisen parameters:

The determination of the mode Grüneisen parameters was achieved by computing the phonon spectra at two additional unit cell volumes, which were deformed from equilibrium volume by expanding or compressing by 1%, and re-optimizing the atomic positions.

In this work, we also predict the dependence of thermal conductivity on temperature by employing the Slack's expression:^19,20

where M̄ [in g mol⁻¹] is the average mass per atom in the cell, δ [m] is the cube root of the average volume per atom, n is the number of atoms in the primitive unit cell, γ(T) represents the macroscopic Grüneisen parameters, computed by eqn (3). Julian²¹ determined the value for the physical constant A [W mol g⁻¹ m⁻² K⁻³] with A = 2.43 × 10⁴/(1 − 0.514/γ + 0.228/γ²). Θ_D [K] is the Debye temperature which can be calculated from the elastic constants.

Besides the phonon spectra, all other properties were calculated using our own home-made post-processing programs.

3. Results and discussions

3.1. Phonon properties and Grüneisen parameters

The phonon dispersion curves give a criterion for the crystal stability. If all phonon frequencies are positive, the crystal is dynamically stable, while the imaginary frequencies (soft modes) indicates the system is unstable with respect to certain atomic displacement patterns. The phonon spectra of both aluminum silicon carbides at equilibrium states are shown in Fig. 1a and 2a. No imaginary frequencies can be found in the whole Brillouin zone for both phases implying their stability nature. We also checked that they remain stable under certain lattice deformation (±10%). In addition, due to the fact that carbon atom has a smaller mass than aluminum and silicon atoms, the vibration frequency of carbon atoms is higher than that of other atoms. As a consequence, the PHDOS of both ternary carbides can be viewed as two parts (Fig. 1b and 2b). One refers to the part lower than 13 THz where the main contribution comes from the Al sub-lattice while the other part higher than 13 THz is dominated by the dynamics of carbon atoms. Both parts have contributions from silicon atoms. From the phonon spectra, a pseudo-gap is seen around 20 THz for two structures. Above this pseudo-gap, lattice vibrations are dominated by C atoms.

Fig. 1 The phonon dispersions (a), partial phonon density of states per ratio of Al₄ : Si : C₄ (b) and the corresponding phonon mode Grüneisen parameters (c) of Al₄SiC₄. The color map is consistent with the increasing of phonon frequency.

Fig. 2 The phonon dispersions (a), partial phonon density of states per ratio of Al₄ : Si₂ : C₅ (b) and the corresponding phonon mode Grüneisen parameters (c) of Al₄Si₂C₅. The color map is consistent with the increasing of phonon frequency.

In the Al₄SiC₄ structure, there are 18 atoms (2 f.u.). Therefore, totally 54 phonon branches exist in the phonon spectrum, including 3 acoustic modes and 51 optical modes. Factoring all 54 phonon modes at Γ-point based on their irreducible representations of point group C_6v, we have Γ₃ = E₁ + A₁ for three acoustic phonons and Γ₅₁ = 8A₁ + 9B₂ + 8E₁ + 9E₂ for remaining optical ones. The crystal structure of Al₄Si₂C₅ contains 33 atoms (3 f.u.), giving rise to a total number of 99 phonon branches. Group theory dissociates three acoustic phonon modes into Γ₃ = E_u + A_2u. Similarly, we obtain Γ₉₆ = 5E_u + 5A_2u + 5E_g + 5A_1g + 11G + 11E, where G and E refer to four-fold and two-fold degeneracies. All phonon modes at Γ point given by G and E representations are neither infrared (IR) active nor Raman active in Al₄Si₂C₅ structure. Using the character table of point group D_3d, one can show that G = E_g + E_u and E = A_1g + A_2u. Therefore, G and E are obtained by a linear combination of IR and Raman active irreducible representations. In Table 1, we summarize the frequencies and irreducible representations of IR and Raman active phonon modes for Al₄SiC₄ and Al₄Si₂C₅ structures.

Table 1 Optical active phonon modes at Γ-point of Al₄SiC₄ and Al₄Si₂C₅. Phonon frequencies, irreducible representations and types of eigenvector are shown. Unit of frequency is cm⁻¹. Numbers between 0 and 1 are indicated as x, y and z

Symmetry	Al4SiC4			Symmetry	Al4Si2C5
	IR	Raman	[u v w]		IR	Raman	[u v w]
E2		78.0	[x y 0]	Eu	118.3		[x y 0]
E2		96.0	[x 0 0]/[0 y 0]	Eg		161.1	[x y 0]
E1		139.7	[x 0 0]/[0 y 0]	Eu	218.7		[x y 0]
E1	188.6	188.6	[x y 0]	A2u	247.3		[0 0 z]
E2		203.9	[x y 0]	A1g		258.5	[0 0 z]
E2		262.6	[x y 0]	Eg		270.4	[x y 0]
E1	268.1	268.1	[x y 0]	Eg		279.0	[x y 0]
E1	278.2	278.2	[x y 0]	A1g		369.4	[0 0 z]
E2		280.5	[x y 0]	A2u	431.9		[0 0 z]
A1	290.1	290.1	[0 0 z]	Eu	476.1		[x y 0]
A1	303.4	303.4	[0 0 z]	A1g		543.5	[0 0 z]
A1	397.9	397.9	[0 0 z]	A2u	561.6		[0 0 z]
E1	443.0	443.0	[x y 0]	Eu	614.6		[x y 0]
E2	443.6	443.6	[x y 0]	Eg		616.0	[x y 0]
A1	517.8	517.8	[0 0 z]	A2u	740.9		[0 0 z]
A1	578.4	578.4	[0 0 z]	Eg		800.4	[x y 0]
E1	596.4	596.4	[x y 0]	Eu	807.1		[x y 0]
E2		596.8	[x y 0]	A1g		810.6	[0 0 z]
A1	720.1	720.1	[0 0 z]	A2u	815.3		[0 0 z]
E1	755.7	755.7	[x y 0]	A1g		880.3	[0 0 z]
E2	755.7	755.7	[x y 0]
E1	777.6	777.6	[x y 0]
E2	777.7	777.7	[x y 0]
A1	807.7	807.7	[0 0 z]
A1	862.0	862.0	[0 0 z]

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The general forms of eigenvector of all optical active phonon modes are also given in Table 1. Phonon eigenvector is related to atomic displacement in a particular phonon mode by the following expression.

u_j(r_i,t) = u⃑_j,0 exp[i(kr − ωt)] (5)

where, k represents wave vector in the first Brillouin zone; r refers to the atomic position; the phonon angular frequency is given by ω; t denotes time and phonon eigenvector u⃑_j,0. Assuming t = 0, then we may rewrite eqn (5) as

u_j(r_i,t) = u⃑_j,0 exp[ikr] (6)

Eqn (6) implies that the atomic displacement is a complex number at an arbitrary k point except the Γ-point. In our discussion, we only consider all phonon modes at Γ-point. Therefore, the phonon eigenvector also indicates the magnitude and direction of atomic displacement directly.

In both Al₄SiC₄ and Al₄Si₂C₅ phases, there are only two types of eigenvector, i.e., [x y 0] and [0 0 z]. Those phonon modes with displacements involving only x–y basal plane are represented by [x y 0] type eigenvector. Meanwhile, other phonon modes moving merely in [001] direction are described by [0 0 z] type eigenvectors. Some representative phonon eigenvectors are illustrated in Fig. 3 and 4 using VESTA software.²² As can be seen from both figures, the in-phase displacement patterns on x–y crystallographic plane correspond to the lower part of phonon spectrum. The out-of-phase motions of Si, Al and C atoms on the same plane have relatively large frequencies, partially because Si–C or Al–C bonds are stretched indirectly. Obviously, vibrations in [001] direction are anticipated to give large phonon frequencies. In Al₄SiC₄ and Al₄Si₂C₅ phases, some Al–C and Si–C bonds are aligned in c-direction. Therefore, displacing atoms in [001] direction either in-phase or out-of-phase would be the most effective way to change the bond length aligned in this direction. As a result, those phonon modes have the largest frequencies in two structures. For example, in Al₄SiC₄ structure, A₁ mode (IR + R, 862.03 cm⁻¹): the out-of-phase compression of Si–C bond in [001] direction; A₁ (IR + R, 807.70 cm⁻¹): out-of-phase compression of Al–C bond in [001] direction; A₁ (IR + R, 720.13 cm⁻¹): out-of-phase stretching of Si–C bonds in [001] direction. Similarly, for Al₄Si₂C₅ structure, A_1g (R, 880.28 cm⁻¹): out-of-phase stretching of Si–C bonds in [001] direction; A_1g (R, 814.64 cm⁻¹): out-of-phase compression of Al–C bonds in [001] direction; A_2u (IR, 247.28 cm⁻¹): in-phase stretching of Al–C and Si–C bonds in [001] direction.

Fig. 3 Eigenvectors of some selected optical active phonon modes at Γ-point of Al₄SiC₄. The arrow indicates the direction of atomic displacement, and the length of arrow refers to the module of eigenvector. The largest balls: Al; smallest ones: Si; medium size: C. VESTA software is used for visualization of eigenvectors. (a) E₂ (R), 78.01 cm⁻¹; (b) E₂ (R), 96.02 cm⁻¹; (c) E₁ (IR + R), 188.59 cm⁻¹; (d) E₂ (R), 262.57 cm⁻¹; (e) A₁ (IR + R), 290.05 cm⁻¹; (f) E₂ (IR + R), 443.59 cm⁻¹; (g) A₁ (IR + R), 578.35 cm⁻¹; (h) A₁ (IR + R), 720.13 cm⁻¹; (i) A₁ (IR + R), 807.70 cm⁻¹; (j) A₁ (IR + R), 862.03 cm⁻¹.

Fig. 4 Eigenvectors of some selected optical active phonon modes at Γ-point of Al₄Si₂C₅. The arrow indicates the direction of atomic displacement, and the length of arrow refers to the module of eigenvector. The largest balls: Al; smallest ones: Si; medium size: C. VESTA software is used for visualization of eigenvectors. (a) E_u (IR), 118.32 cm⁻¹; (b) A_2u (IR), 247.28 cm⁻¹; (c) E_u (IR), 476.07 cm⁻¹; (d) E_u (IR), 614.61 cm⁻¹; (e) A_1g (R), 814.64 cm⁻¹; (f) A_1g (R), 880.28 cm⁻¹.

The acoustic branches of the both ternary carbides can be employed to estimate the elastic constants by Christoffel equation.²³

|Γ_ij − ρv²δ_ij| = 0 (7)

where, the density is given by ρ and v represents the group velocity of an acoustic phonon branch in Debye approximation; δ_ij is the Kronecker delta function. Γ_ij is usually refereed to acoustic tensor, and which can be obtained from eqn (8).

The directional cosines are n₁ = sin(θ)cos(φ), n₂ = sin(θ)sin(φ), and n₃ = cos(φ). Solving eqn (8) for hexagonal crystal structure in Γ–A, Γ–M and Γ–K directions, the solutions are related to elastic constants by the following expressions.

In Γ–A direction ([001]),

In Γ–M direction ([010]),

In Γ–K direction [110],

here, v_l and v_t are longitudinal and transverse sound velocities. The solutions in [110] direction are the same as those of [010] direction, as implied by hexagonal symmetry. Using the current phonon spectra, we are able to calculate C₁₁, C₁₂, C₃₃ and C₄₄. The results are tabulated in Table 2.

Table 2 Calculated longitudinal and transverse sound velocities along different directions (km s⁻¹), and second order elastic constants (C_ij in GPa) of Al₄SiC₄ and Al₄Si₂C₅

Compounds	Sound velocities					Elastic constants
	[001] vl	[100] vt	[010] vl	[100] vt1	[001] vt2	C11	C12	C33	C44
a Cal. data by stress-strain method.b Cal. data obtained by Liao et al.^11
Al4SiC4	11.2	6.1	11.0	7.2	5.9	363, (369.5^a, 386^b)	52.1, (117^a, 118^b)	376.3, (383.5^a, 409^b)	111.6, (110.2^a, 122^b)
Al4Si2C5	7.3	6.1	11.1	7.4	5.9	373.3, (371.7^a)	41.5, (112.1^a)	161.5, (396^a)	112.7, (100.5^a)

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Both compounds show slightly anisotropy in sound wave velocity. The sound velocities ranging from 5.9–11.2 km s⁻¹ and 5.9–11.1 km s⁻¹ can be observed for Al₄SiC₄ and Al₄Si₂C₅, respectively. Meanwhile, the calculated elastic constants of Al₄SiC₄ are C₁₁ = 363 GPa, C₁₂ = 52.1 GPa, C₃₃ = 376.3 GPa, C₄₄ = 111.6 GPa; and the values of Al₄Si₂C₅ are C₁₁ = 373.3 GPa, C₁₂ = 41.5 GPa, C₃₃ = 161.5 GPa, C₄₄ = 112.7 GPa. Compared with the results obtained by stress-strain method,^24,25 the values of C₁₁ and C₄₄ are in good agreement between two methods (in Table 2). The biaxial elastic modulus C₁₂ is significantly smaller than that predicted by strain–stress method. In the case of Al₄Si₂C₅, the uniaxial modulus C₃₃ is also underestimated, compared to strain–stress value. Since we employ PBE to calculate phonon spectrum, it is likely that acoustic phonon dispersions in [001] direction are become softer than they would be.

Based on the phonon spectra of the equilibrium and deformed cells (±1%), the mode Grüneisen parameters are obtained, and the results are displayed in Fig. 1c and 2c. For Al₄SiC₄, the mode Grüneisen parameters are positive throughout the Brillouin zone for almost all phonon branches except for few of them exhibit negative values. On the other hand, Al₄Si₂C₅ phase has a large number of phonon modes with negative γ(ω) values, and all of them are seen in both acoustic and optical branches in low frequency part. The observed negative γ(ω) at low frequency part of Al₄Si₂C₅ may be relevant to its thermal expansion behavior, i.e., at very low temperature, the thermal expansion coefficients of Al₄Si₂C₅ may decrease with the increasing of temperature. The origin of negative mode-Grüneisen parameter is illustrated in Fig. 5, using a phonon eigenvector of Al₄SiC₄ as an example. This particular phonon mode at Γ-point gives a negative mode-Grüneisen parameter of −0.28. As can be seen from Fig. 5, the eigenvector is obviously [x y 0] type, and which corresponds to the in-plane movement of all atoms. The eigenvectors of all Si and C atoms are in-phase on basal plane. Out-of-phase motions are seen for Al and C atoms in CAl₄ and CSi₃Al tetrahedral. The most intriguing result due to those atomic displacements is the creation of stacking fault between Si and Al layers. As a result, it is likely to increase the electrostatic repulsion between two positively charged atomic layers due to the lack of proper screening, i.e., Al–Al and Al–Si. Such repulsive interaction is weakened by increasing the cell volume. Therefore, the force constant and phonon frequency are actually larger in a stretched cell than those of a compressed cell. For Al₄SiC₄ and Al₄Si₂C₅, negative mode-Grüneisen parameter is not observed in high-frequency optical phonons. Most lower-laying optical phonons are very likely to create stacking fault on basal plane in the crystal structure. Therefore, we may conclude that the increasing of electrostatic energy between two mismatched atomic layers due to the stacking fault results in the negative mode-Grüneisen constant in both structures.

Fig. 5 The creation of stacking fault in Al₄SiC₄ crystal structure due to a phonon mode (E₁, 139.7 cm⁻¹, Raman active) at Γ-point with negative mode-Grüneisen constant (−0.28). (a) 3-D side-view; (b) 2-D side view. The stacking fault is clearly seen in (b).

The macroscopic Grüneisen parameters are calculated by eqn (3) and the results are shown in Fig. 6. Basically, at low temperature, the macroscopic Grüneisen parameter is determined by acoustic phonons and low-frequency optical band modes; and at high temperature, the macroscopic Grüneisen parameters are dominated mostly by optical phonons. The macroscopic Grüneisen parameters along a or b axis (γ_[100]) increase continuously with the temperature (Fig. 6a and c) for Al₄SiC₄ and Al₄Si₂C₅. However, the γ_[001] versus T curves show different behaviors in both ternary carbides. Here, γ_[001] of Al₄SiC₄ increases with the temperature, and the opposite trend is obtained for Al₄Si₂C₅. This is reasonable because the optical mode-Grüneisen parameters of Al₄SiC₄ have larger values along H–K and M–L paths ([001] crystal directions) than those of Al₄Si₂C₅. Meanwhile, the Grüneisen parameter of both phases along c axis is higher than that along a or b axis. The observed anisotropy in Grüneisen parameter reflects the difference in chemical bonding of [001] direction and x–y basal plane. As a result, we would expect the linear thermal expansion coefficient of [001] direction will be higher than that along [100] or [010] direction.

Fig. 6 The macroscopic Grüneisen parameters for Al₄SiC₄ (a and b) and Al₄Si₂C₅ (c and d) in principal directions.

3.2. Thermodynamic properties

The obtained Helmholtz free energy curves for temperatures ranging from 0 K to 1600 K are shown in Fig. 7. We can clearly see that the free energy decreases with the temperature, and the equilibrium volume increases with the temperature. Based on this facts, the volumetric and linear thermal expansion coefficients are shown in Fig. 8. For hexagonal structure, the linear TECs can be calculated from volumetric TECs and macroscopic Grüneisen parameters:¹⁸

λ(T) = 2α_a(T) + α_c(T) (12)

where the ratio of mean Grüneisen constant γ_[100](T)/γ_[001](T) is computed from the flat region in Fig. 6 at high temperature. The calculated volumetric and linear TECs for aluminum silicon carbides are shown in Fig. 8. The TECs of Al₄SiC₄ is higher than that of Al₄Si₂C₅ in temperature range 0–1600 K. At room temperature, the calculated average linear TECs (simplified as α = λ/3) for Al₄SiC₄ and Al₄Si₂C₅ are 8.9 × 10⁻⁶ K⁻¹ and 7.9 × 10⁻⁶ K⁻¹, respectively. At low temperature, the TECs of both ternary carbides increase rapidly, which is mainly caused by the increasing of lattice vibration energy. Above room temperature, the TECs increase linearly with temperature. As expected in last section, the linear TECs along c axis (α_c) are slightly higher than that along a axis (α_a). In ref. 5, Wen and Huang determined the experimental linear TEC of Al₄SiC₄, they gave a average value of 6.2 × 10⁻⁶ K⁻¹ in the range 473–1723 K. Wills and Goodrich²⁶ claimed the experimental linear TEC of Al₄SiC₄ is 6.9 × 10⁻⁶ K⁻¹ without indicating the actual temperature. Recently, Inoue et al.⁶ provided an average experimental value of 7.16 × 10⁻⁶ K⁻¹ for highly densified Al₄SiC₄ between 300 K and 1473 K. In this work, the temperature averaged linear TECs for Al₄SiC₄ and Al₄Si₂C₅ in the range from 300 K to 1500 K are 9.96 × 10⁻⁶ K⁻¹ and 8.9 × 10⁻⁶ K⁻¹, respectively. The obtained linear TECs of Al₄SiC₄ is slightly higher than the experimental data. There are two possible reasons for the discrepancies. Firstly, as we know, the equilibrium volume at each temperature is crucial for calculating TECs. When the cell volume is overestimated by PBE functional, the force constants might be underestimated. Therefore, the phonon frequencies probably are underestimated, compared to their actual values. As a result, TECs could be larger than experiments using QHA. Besides, we should note that the calculations are performed for the perfect crystal structure. While for experimental ternary carbide ceramics, there are defects in the specimens such as pores and impurities. They may lead to smaller TECs in experimental data. We would like to list the average linear TECs of several other MAX phases like Ti₃SiC₂ (10 × 10⁻⁶ K⁻¹),²⁷ Ti₃AlC₂ (9.2 × 10⁻⁶ K⁻¹),²⁸ Ti₂AlC (9.2 × 10⁻⁶ K⁻¹),²⁹ Ti₅Al₂C₃ (9.3 × 10⁻⁶ K⁻¹).²⁹ Our linear TEC values of Al₄SiC₄ and Al₄Si₂C₅ are comparable with these data.

Fig. 7 Dependence of the Helmholtz free energy F (V, T) on the cell volume at different temperature from 0 to 1600 K for Al₄SiC₄ (a) and Al₄Si₂C₅ (b). The calculated equilibrium volumes of Al₄SiC₄ and Al₄Si₂C₅ at 0 K are 204.45 ų and 369.09 ų, respectively. These values are slightly higher than experimental data (201.6 ų for Al₄SiC₄ and 367.1 ų for Al₄Si₂C₅) obtained by Inoue et al.¹ Because the PBE functional was used for calculation.

Fig. 8 Thermal expansion coefficients (TECs) of Al₄SiC₄ (a) and Al₄Si₂C₅ (b) as a function of temperature.

Using eqn (11) and (12), the linear TEC tensor can be determined from volumetric TEC.¹⁸ The calculated tensor components are used here to illustrate the anisotropy in TEC of hexagonal Al₄SiC₄ and Al₄Si₂C₅. In Fig. 9, the two-dimensional planar projections are shown for two structures. In the calculations, we use temperature-averaged linear TECs computed between 300 K and 1500 K. For Al₄SiC₄, we find 9.75 × 10⁻⁶ K⁻¹ in [100] direction and 11.1 × 10⁻⁶ K⁻¹ in [001] direction, and the values for Al₄Si₂C₅ in those directions are 8.79 × 10⁻⁶ K⁻¹ and 9.69 × 10⁻⁶ K⁻¹. For both structures, the linear TECs in two principal crystallographic directions are not significantly different to each other. The anisotropy in linear TEC tensor is expected, as shown in Fig. 9. However, Al₄SiC₄ and Al₄Si₂C₅ are not considered as the typical layered structures where physical properties are very different in the directions parallel and perpendicular to layers in latter case. From our current results, the chemical bonds intra- and inter-layer are not quite unlike in two structures.

Fig. 9 Anisotropy in thermal expansion tensor is illustrated by 2-D projection. (a) Al₄SiC₄; (b) Al₄Si₂C₅.

Isothermal bulk modulus as a function of temperature is shown in Fig. 10. Al₄Si₂C₅ has larger bulk modulus than Al₄SiC₄ implying the small incompressibility of Al₄Si₂C₅. We may conclude that incorporating more Si and C atoms in the Al–Si–C ternary compounds would increase the mechanical modulus, because the formation of more Si–C bonds in the crystal structure of Al₄Si₂C₅. The bulk moduli of both phases decrease with increasing of temperature. At 1600 K, the bulk modulus of Al₄SiC₄ is decreased by ∼33.8 GPa of the value at 0 K, and which is little smaller than that of Al₄Si₂C₅ by ∼34.9 GPa. Then, isobaric bulk modulus as a function of temperature can be computed using expression below:

where K_S(T) is the adiabatic bulk modulus, and K_T(T) refers to the isothermal bulk modulus. The calculated results are shown in Fig. 10b. At very low temperature, the difference between isobaric bulk modulus and isothermal bulk modulus is small (C_P and C_V are almost the same). While with the increasing of temperature, the isobaric bulk modulus decreases much more slowly than isothermal bulk modulus due to thermal expansion. In our case, the difference of K_S(T) and K_T(T) between 0 K and 1600 K is no larger than 10 GPa.

Fig. 10 Isothermal bulk modulus (a) and isobaric bulk modulus (b) of aluminum silicon carbides as a function of temperature.

Base on the thermal expansion results, we calculated the specific heat at constant pressure (C_P) with C_P = C_V + λ²V(T)TK_T.²³ Here, λ is the volumetric TEC, K_T is the isothermal bulk modulus at different temperature. Meanwhile, the specific heat C_V and entropy S as a function of temperature were evaluated by standard expressions in ref. 30 and 31.

The specific heats (C_V and C_P) results are illustrated in Fig. 11. The obtained C_P values of Al₄SiC₄ and Al₄Si₂C₅ at room temperature are 16.5 and 18.8 J mol⁻¹ K⁻¹·atom, respectively. Constant pressure specific heat increases with temperature due to volumetric thermal expansion. Above room temperature, an increasing of the value linearly with temperature implies that anharmonic effects are relatively weak in crystal structure. Thus, QHA is an accurate approximation for computing thermal properties. Below room temperature, the difference between C_P and C_V is very small because the thermal expansion is nearly negligible. C_P and C_V behave differently at high temperature, C_V tends to approaching the classical Dulong–Petit asymptotic limit of 24.93 J mol⁻¹ K⁻¹; while C_P keeps growing at high temperature due to the work done by the volumetric expansion.

Fig. 11 The specific heats at constant pressure or volume of Al₄SiC₄ (a) and Al₄Si₂C₅ (b) as a function of temperature. The horizontal line represents classic Dulong–Petit law.

In Fig. 12 the dependence of the entropy on the temperature of both phases are shown. The calculated values for Al₄SiC₄ and Al₄Si₂C₅ at 300 K are 12.2 and 11.7 J mol⁻¹ K⁻¹·atom and increase continuously with the temperature. Al₄Si₂C₅ has a slightly higher value than Al₄SiC₄ due to more vibrational modes in former structure.

Fig. 12 The entropy of aluminum silicon carbides as a function of temperature.

The thermal conductivity, which characterizes the heat transportation capability of crystal structures, are very important for high-temperature structural materials. The dependence of thermal conductivity on temperature is plotted in Fig. 13. As expected from Slack model, thermal conductivity decreases with the increasing of temperature by the law 1/T. The average thermal conductivities in the temperature range of 300–1500 K are 29.9 and 23.4 W m⁻¹·K⁻¹ for Al₄SiC₄ and Al₄Si₂C₅, respectively. The unit cell of Al₄Si₂C₅ has more atoms than formal structure, resulting in a lower thermal conductivity due to more optical phonons in the structure.³² For comparison, we want to list some available experimental data for Al₄SiC₄. Wills and Goodrich²⁶ gave the experimental thermal conductivity of 8 W m⁻¹ K⁻¹ without clearly indicating the temperature. Inoue et al.⁶ claimed a constant thermal conductivity of 80 W m⁻¹ K⁻¹ in the temperature range from 300 K to 1473 K. Actually, our calculated average thermal conductivity of Al₄SiC₄ is situated between them. Unfortunately, to our knowledge, the experimental thermal conductivity for Al₄Si₂C₅ is not available at the moment.

Fig. 13 Thermal conductivities (k) of aluminum silicon carbides as a function of temperature.

3.3. Chemical bonding

In this part, we would like to discuss the chemical bonding properties of the aluminum silicon carbides under hydrostatic pressure. Fig. 14 shows the normalized bond lengths as a function of pressure. Generally speaking, all bond lengths decrease with the increasing of pressure. The chemical bonds, which are easy to change under pressure, are situated on the lower part of graphs, showing a stronger dependence on pressure. Specifically, Al–C bonds are softer than Si–C bonds under compression. In both Al₄SiC₄ and Al₄Si₂C₅ structures, Si–C bonds aligned in [001] direction are stiff and exhibit small compressibility among all chemical bonds. Although, several different bond lengths are observed for either Si–C or Al–C bonds, the general trend is the same in two structures.

Fig. 14 The normalized bond lengths of Al₄SiC₄ (a) and Al₄Si₂C₅ (b) as a function of pressure.

Using the volume at different pressures, we could calculate the isothermal bulk modulus by fitting volume versus pressure data, using a proper equation of state (EOS). The bulk moduli of Al₄SiC₄ and Al₄Si₂C₅ are 173.6 and 178.2 GPa, and which are slightly higher than the data shown in Fig. 10. The uniaxial modulus can be defined by B_x = −x₀(dP(x)/dx) where x refers to lattice parameters (a or c). The calculated uniaxial moduli of the two ternary carbides show anisotropy. For Al₄SiC₄ the B_a and B_c are 541.4 and 478.7 GPa, while for Al₄Si₂C₅, the values are 558.7 and 489.8 GPa, respectively. These uniaxial moduli along principal axes are expected to be at least comparable to elastic constants in the same direction. The calculated values are actually larger than the corresponding elastic constants. The observed discrepancies in the two data sets can be traced back to the calculation of elastic constants under finite strain (hydrostatic pressure), as shown in eqn (15).

where, C_ijkl is elastic constants obtained by normal means, i.e., the derivative of Piola-Kirchhoff stress tensor with respect to strain (deformation gradient); meanwhile, σ_ij refers to symmetric stress tensor (For example, second order Piola-Kirchhoff stress tensor). It is easy to illustrate that the mechanical moduli obtained from eqn (15) at finite strain are larger than those merely obtained from C_ijkl only.

The value of B_a is larger than that of B_c implying the anisotropy in compressibility. In addition, the difference between B_a and B_c is mainly attributed to layered like crystal structure in MAX phases.

The insightful chemical bonding information are provided by population analysis results shown in Table 3. It is clearly indicated that C carries the negative charges in both ternary carbides. The positive charges are carried by Al and Si atoms. We suggest two possible electron transfer mechanisms. The first one refers to s–p orbital hybridization of the same atom (C, Al and Si). The other one is induced by the ionic bonding among C and Al (or Si). In the former case, the electrons are transferred from (C, Al, Si)-s states to (C, Al, Si)-p states of same atom, and those hybridized atomic orbitals overlap with each other in upper valence band region between atoms. In the latter case, the electrons are transferred from Al atom to C atom, implying an ionic bonding character between them. However, the bonding mechanism between Si and C is anticipated to be mainly covalent. The calculated bond overlap populations and bond lengths of Si–C pair confirm this conjecture. Besides, for both ternary carbides, the Al–C (and Si–C) bond deviate from c axis behave the largest overlap populations and relatively short bond length, compared to those of Al–C bonds aligned along c axis. For Al–C bond along c axis type II, a quite small electron-overlap population value (0.08 electrons) is found, and which implies Al–C bond along [001] direction is close to perfect ionic bond. The quite large bond length also supports this conclusion. From the calculated linear bulk moduli in [100] and [001] directions, we may conclude that the overall bond strength at basal plane (x–y) is slightly stronger than that of z direction in two hexagonal structures.

Table 3 Population analysis results of Al₄SiC₄ and Al₄Si₂C₅, bond length (L, in Å) and the average populations (n, electrons) are also given by Mulliken's method

Polymorph	Species (ion)	s	p	Total electrons	Charges
Al4SiC4	C(1–8)	1.49	3.92	5.41	−1.41
	Al(1–8)	0.66	1.2	1.86	1.14
	Si(1–2)	1.04	1.84	2.88	1.12
Al4Si2C5	C(1–15)	1.48	3.91	5.39	−1.39
	Al(1–12)	0.67	1.17	1.84	1.16
	Si(1–6)	1.03	1.84	2.87	1.13

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Polymorph	Bond	n	L
Al4SiC4	Al–C bond along c axis type I	0.45	1.91
	Al–C bond along c axis type II	0.08	2.18
	Al–C bond along c axis type III	0.44	1.96
	Al–C bond deviate from c axis	Avg. 1.68, (1.37–2.00)	Avg. 2.03, (1.92–2.14)
	Si–C bond along c axis	0.49	1.86
	Si–C bond deviate from c axis	2.25	1.98
Al4Si2C5	Al–C bond along c axis type I	0.45	1.95
	Al–C bond along c axis type II	0.08	2.21
	Al–C bond deviate from c axis	Avg. 1.65, (1.36–1.94)	Avg. 2.02, (1.91–2.13)
	Si–C bond along c axis	0.5	1.86
	Si–C bond deviate from c axis	2.25	1.97

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Finally, we would like to analyze the charge density of Al₄SiC₄ structure at the equilibrium volume. Similar plots for Al₄Si₂C₅ are omitted here, because they show little difference to Al₄SiC₄ case. In Fig. 15, the total electron density is shown on (100) crystallographic plane. The accumulation of charge density around different atoms is consistent with their electronegativity values. The average electron density at the position of C atom is significantly higher than those of Al and Si. This observation is supported by the electron population results given in Table 3. Obviously, electron density of Al–C bond is strongly polarized, and the ionic bond between them is further confirmed. Meanwhile, Si–C bond shows covalent character, as implied by large charge density shared between two atoms. The local uniformity of electron density in the system is depicted in Fig. 15, using reduced density gradient (s). We use the eqn (15) to calculate s from total electron density on a crystallographic plane.

where, n is the electron density, and ∇_xn(x, y) and ∇_yn(x, y) represent the density gradient in x and y direction. For the uniform electron gas, we have s = 0. In the case of a typical metallic structure, the delocalized electron density in the interstitial region usually has small s value. For non-polar covalent bond, s is also close to zero in the direction of bond axis. However, the structure with strong polar bonds exhibits large s between atoms.³³ Our results for Al₄SiC₄ and Al₄Si₂C₅ indicate a highly non-uniform electron density distribution both locally and globally due to polar bonds such as Al–C and Si–C. We also notice that s is relatively small in certain region between C and Si atoms, and which implies Si–C bond is mainly covalent.

Fig. 15 Charge density properties of Al₄SiC₄. (a) Electron density distribution at (100) plane; (b) reduced density gradient (s) at the same plane. The unit for electron density is e Å⁻³.

4. Conclusions

Using first-principles calculations based on DFT, the phonon spectra and Grüneisen parameters were calculated. No imaginary phonon frequencies can be found in the whole Brillouin zone for Al₄SiC₄ and Al₄Si₂C₅ implying their stability nature. IR and Raman active phonon modes and their eigenvectors were analyzed. The Grüneisen parameters of both phases along c axis are higher than that along a or b axis. Therefore, the linear TECs of both ternary carbides along [001] direction are higher that along [100] or [010] direction. The average linear TECs for Al₄SiC₄ and Al₄Si₂C₅ in the temperature range from 300 K to 1500 K are 9.96 × 10⁻⁶ and 8.9 × 10⁻⁶ K⁻¹, respectively. The calculated components of TEC tensor are used to illustrate its anisotropy in Al₄SiC₄ and Al₄Si₂C₅. The results indicate both compounds are not considered as the typical layered structures where physical properties are very different in the directions parallel and perpendicular to layers. The C_P of Al₄SiC₄ and Al₄Si₂C₅ at room temperature are 16.5 and 18.8 J mol⁻¹ K⁻¹·atom and increase monotonically with temperature up to 1600 K. The average thermal conductivities in the temperature between 300 K and 1500 K are 29.9 and 23.4 W m⁻¹ K⁻¹ for Al₄SiC₄ and Al₄Si₂C₅, respectively. The uni-axial modulus B_a and B_c of Al₄SiC₄ are 541.4 and 478.7 GPa, while for Al₄Si₂C₅, the values are 558.7 and 489.8 GPa, respectively. Chemical bonding properties were investigated under different hydrostatic pressures by Milliken population analysis. The electron density and its reduced density gradient distributions on (110) crystallographic plane were shown to illustrate the characteristics of Al–C and Si–C bonds in the structures. It was found that Si–C bonds are mainly covalent, in contrast to the ionic dominated Al–C bonds.

Acknowledgements

This work was supported by the Natural Science Foundation of China (51501139), the Science and Technology Project of Guangdong Province in China (Special Foundation for Additive Manufacturing Technologies: No. 2015B010122003, and Special Foundation for Practical Science and Technology Research and Development Project: No. 2015B090926009), the Postdoctoral Science Foundation funded project of China (No. 2014M552434).

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