First-principles investigation into the effect of pressure on structural, electronic, elastic, elastic anisotropy, thermoelectric and thermodynamic properties of CaMgSi

Results in Physics 14 (2019) 102483

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First-principles investigation into the effect of pressure on structural, electronic, elastic, elastic anisotropy, thermoelectric and thermodynamic properties of CaMgSi

T

⁎

Wen-Duo Han, Ke Li , Jia Dai, Yao-Hui Li, Wen-Lai Yin School of Mechanical and Electrical Engineering, Nanchang University, China

A B S T R A C T

In this work, the structure, electronic, elastic, elastic anisotropy, thermoelectric, and thermodynamic properties of CaMgSi under pressure are calculated via firstprinciples methods. The optimized structure and lattice parameters are in agreement with previously published experimental results, indicating the methods used in this work are feasible and reliable. The calculated results show that the values of bulk modulus, B, Young’s modulus, E, and shear modulus, G, increase with the pressure overall, while the values of G and Y decrease in special pressure regimes from 20 to 24 GPa. Analyses of Poisson’s ratio, v, and Pugh’s modulus ratio, B/G, imply that CaMgSi is prone to change from brittle to ductile with increasing pressure. The transition from brittleness to ductility occurs at a pressure of 16 GPa. The value of Hv decreases with the pressure overall, although it increases over the pressure ranges of 16–20 GPa and 24–28 GPa. Universal elastic anisotropy increases at pressures from 4 to 24 GPa, while it decreases over the pressure ranges from 0–4 and from 24–28 GPa. Moreover, the variation of elastic anisotropy for G and E is larger compared to that of B. In addition, the electrical conductivity and electronic thermal conductivity decrease with increasing pressure based on the total density of states for CaMgSi. In addition, the heat capacity decreases with increasing pressure, indicating the reduction of lattice thermal conductivity, which is beneficial for thermoelectric performance, especially at pressures from 8 to 16 GPa. Meanwhile, the structural stability can be enhanced at higher pressures, while the thermal stability of crystalline CaMgSi becomes weaker with increasing pressure. The investigation presented in this work offers a comprehensive understanding of the effects of pressure on the mechanical, thermoelectric, and thermodynamic properties of CaMgSi, which can provide guidance for further theoretical work and practical applications.

Introduction Over the past few decades, the development of new sustainable and eco-friendly energy sources has become an increasingly urgent task due to the severe threats of energy shortages and environmental pollution. A great deal of attention has been focused on thermoelectric devices for their unique function of converting thermal power into available electrical energy without any additional mechanical energy, and the key part of a thermoelectric device is the thermoelectric material. Conventional thermoelectric materials of the form X2Te3 (X = Be, Sb, Pb) [1–3], which exhibit favorable efficiency of conversion (ZT values), are of limited use in practice due to their toxicity and high cost. Therefore, it is important to develop cheaper, environmentally benign, and eco-friendly thermoelectric materials. Mg2Si-based compounds, which have lower density (1880 kg/m3) [4] compared to X2Te3 (X = Be, Sb, Pb) compounds (7860 kg/m3) [5], have attracted significant attention due to their favorable thermoelectric and eco-friendly performance [6,4,7–10]. Recently, CaMgSi compounds were synthesized by Miyazaki et al. [11] and they were found to exhibit remarkable thermoelectric performance. Profiting from

⁎

the low density, nontoxic nature, and low cost of CaMgSi compounds, much research has been dedicated to examining their crystal structures as well as their electronic and thermoelectric properties. Hidetoshi Miyazaki et al. [12] confirmed that CaMgSi possesses a band gap of 0.26 eV with a semiconductor-like electronic structure. The thermoelectric properties and dynamical stability of CaMgSi intermetallic compounds were predicted by Jianhui Yang et al. via ab initio methods and Boltzmann transport theory [13]. In general, it is important to obtain the physical properties, including structure, elasticity, elastic anisotropy, and thermal stability when a new material is synthesized and applied in practice [14]. The elastic moduli, including Young's modulus, bulk modulus, and shear modulus, can be employed to characterize the mechanical performance of a new material [15]. Moreover, elastic anisotropy can help to determine the preferential direction under pressing, which is affected by various bond strengths along different crystal axes [16,17]. Meanwhile, the thermal conductivity has a close relationship with the anisotropic chemical bonds [18,19]. Generally, low thermal conductivity is a crucial property for the application of a thermoelectric material. However, at present, little research has been dedicated to

Corresponding author at: School of Mechanical and Electrical Engineering, Nanchang University, 999 Xuefu Road, Honggutan District, Nanchang, 330031, China. E-mail address: [email protected] (K. Li).

https://doi.org/10.1016/j.rinp.2019.102483 Received 14 May 2019; Received in revised form 25 June 2019; Accepted 25 June 2019 Available online 03 July 2019 2211-3797/ © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

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(a)

(b)

(c)

Fig. 1. (a) Equilibrium structure of CaMgSi, with Ca, Mg, and Si atoms represented by green, red, and blue spheres, respectively. (b) Variation of lattice parameter with the applied pressure and (c) ratio of X/X0 (X = a, b, and c). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Methods and computational details

Table 1 Experimental and calculated lattice parameters (a, b, and c) and energy of CaMgSi. Pressure (GPa)

a

b

c

V (Å3)

Energy (eV)

Ref

0

7.496 7.460 7.475 7.297 7.139 7.002 6.880 6.766 6.648 6.539

4.440 4.443 4.427 4.350 4.290 4.235 4.214 4.184 4.160 4.140

8.366 8.296 8.315 8.123 7.960 7.852 7.699 7.601 7.527 7.458

278.425 274.969 275.159 257.871 243.825 232.826 223.201 215.220 208.189 201.948

−8336.5 – – −8329.9 −8323.6 −8317.7 −8312.0 −8306.5 −8301.2 −8296.1

Present work [2324]

4 8 12 16 20 24 28

Present Present Present Present Present Present Present

All the calculations were performed with the CASTEP software package based on density functional theory [21]. The elastic tensors were further analyzed after the CaMgSi crystal structures were relaxed to their equilibrium geometries. The elastic constants are described based on the stress–strain relationship with the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) [22]. The cutoff energy of the electronic wave functions was set to 340 eV in reciprocal space. The valence electron configurations applied for computation were Ca 3s23p6 4s2, Mg 2p6 3s2, and Si 3s2 3p2. A 3 × 6 × 3 kpoint mesh was used to perform the Brillouin zone integrations. The structural degrees of freedom were relaxed after the mean Hellmann–Feynman force on each atom converged to 0.03 eV/Å. The total energy convergence was set to 1.0 × 10−5 eV/atom, the maximum strain amplitude was set as 0.05 GPa, and the value of maximum displacement was 1.0 × 10−5 eV/atom.

work work work work work work work

determining the structure, electronic, elastic anisotropic, and thermodynamic properties of CaMgSi under pressure. The microstructure and physical properties of this material can be influenced by pressure [20]. Therefore, we have undertaken this study to systematically and deeply investigate the physical properties of CaMgSi in order to provide significant guidance for its future application. In this work, the structure, elastic, electronic, and thermodynamic properties of CaMgSi are calculated at pressures ranging from 0 to 28 GPa via a first-principles method based on density functional theory (DFT).

Results and discussions Structure of CaMgSi Crystalline CaMgSi is an orthorhombic ternary compound that belongs to the space group op12 (NO. 62). There are 12 atoms in each unit cell. The positions of Ca, Mg, and Si in the CaMgSi compound are 4c (0.5195, 0.250, 0.6790), 4c (0.6431, 0.250, 0.0625), and (0.2701, 2

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(a)

(b)

Fig. 2. (a) The calculated elastic constants and (b) elastic modulus for CaMgSi.

Fig. 3. Pressure dependence of elastic modulus Cauchy pressure, Pugh's ratio, anisotropy, Poisson's ratio, and hardness for CaMgSi.

0.250, 0.1148), respectively. The crystal structure of a CaMgSi unit cell is shown in Fig. 1. The unit cell is composed of four atoms each of Ca, Mg, and Si. Table 1 lists the optimized lattice parameters and the data published in previous works. The lattice parameters optimized in present work agree well with the results presented in the literature from theory and experiment [23,24], within an acceptable deviation of 1%, indicating that the calculations are feasible and reliable. Based on the Birch–Murnaghan equation of state (EOS) [25], the pressure–volume (P–V) curve can be obtained by fitting pressure and volume data, and the fitted curve is shown in Fig. 1(b). Meanwhile, the variations in the lattice parameters (a/a0, b/b0, c/c0, and v/v0) under external pressure are shown in Fig. 1(c). Obviously, the lattice parameters decrease with increasing external pressure. Moreover, the variance rate of a/a0 with pressure is the largest while that of b/b0 is the smallest, indicating that it is easier to compress a CaMgSi crystal along the b-axis while it is more difficult along the a-axis. The equilibrium lattice parameters a0, b0 and c0, the bulk modulus B0, and its pressure derivative B0′ can be obtained by fitting P–V and X/X0–P (X = a, b, and

Fig. 4. Universal elastic anisotropic index AU, percent anisotropy percent anisotropy in bulk (AB), shear modulus (AG), and Young’s modulus (AE).

c) curves as follows:

P (V ) =

V V0

5 3

⎩

3 (B0′ 4

− 4) ⎡ ⎢ ⎣

( ) V V0

2 3

⎫ − 1⎤ ⎥⎬ ⎦⎭

(1)

a a0 = 5.17113 × 10−5P 2 − 0.00588P + 0.99771

(2)

b b0 = 7.51488 × 10−5P 2 − 0.00439P + 0.99792

(3)

c c0 = 8.63095 × 3

7 3

⎧ −( ) ⎤ ( ) ⎥ ⎨1 + ⎣ ⎦

3B0 ⎡ V 2 ⎢ V0

10−5P 2

− 0.00619P + 0.0. 9975

(4)

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(a)

(b)

(d)

(c)

Fig. 5. The surface construction of Young's modulus at (a) 0 GPa, (b) 8 GPa, (c)16 GPa, and (d) 24 GPa.

V V0 = 2.69345 × 10−5P 2 − 0.01661P + 0.99483

(5)

addition, the value of the formation enthalpy can be employed to evaluate the alloying ability. In general, a negative formation enthalpy indicates an exothermic process, implying a phase can form and exist stably [28]. The formation enthalpy of the CaMgSi phase was obtained based on the following formula [29]:

where V0 and B0 are the volume and bulk modulus under zero external pressure, respectively, and B0′ represents the pressure derivative. Moreover, the reduction rates of a/a0, b/b0, c/c0, and v/v0 are 12.8%, 6.8%, 10.9%, and 27.5% under a pressure of 28 GPa, respectively. However, the variations in the lattice parameters decrease with increasing applied pressure due to the greater mutual repulsion between the atoms, leading to increased difficulty of deformation with increasing pressure.

ΔH =

Theoretically, the structural stability of a material can be evaluated based on its cohesive energy. Generally, a negative value of the cohesive energy corresponds to a stable structure [26]. In this work, the cohesive energies of the CaMgSi phase at 0 GPa were obtained according to the following function [27]:

1 A B c (Etot − xEatom − yEatom − zEatom ) x+y+z

(7)

A where the energies for single Ca, Mg, and Si atoms in bulk states, Esolid , C B E Esolid , and solid , are calculated as −998.3, −974.1, and −107.3 eV/ atom, respectively. The calculated formation enthalpy is −11.4 eV, indicating that a CaMgSi phase can form and exist stably.

Cohesive energies and formation enthalpies

Ecoh =

1 A B C (Etot − xEsolid ) − yEsolid − zEsolid x+y+z

Elastic properties The investigation of the single-crystal elasticity is widely employed to evaluate the elastic properties of materials in practice. The singlecrystal elastic constants are calculated based on the stress–strain approach. The single-crystal constants were calculated under pressures ranging from 0 to 28 GPa. The elastic parameters of CaMgSi, such as Cij, B, G, and E, which are shown in Fig. 2(a), can be employed to evaluate the elastic properties. Based on the Born–Huang theory of lattice dynamics, the mechanical stability of orthorhombic CaMgSi under applied pressure can be obtained [30]. The calculated single elastic constants must satisfy the mechanical stability criteria for an orthorhombic structure [31]:

(6)

where Etot is the total energy in each unit cell, and x, y, and z represent the numbers of Ca, Mg, and Si atoms in each unit of the CaMgSi phase, C B A , Eatom , and Eatom are the total energies of single respectively. Here, Eatom atoms of Ca, Mg, and Si in a free state, respectively. The calculated energies of Ca, Mg, and Si single atoms are −998.1 eV/atom, −972.5 eV/atom, and −102.1 eV/atom, respectively. The calculated value of the cohesive energy is −4.4 eV, which is below zero, indicating that CaMgSi is structurally stable at 0 GPa. In 4

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(b)

(a)

(d)

(c)

Fig. 6. The surface construction of shear modulus at (a) 0 GPa, (b) 8 GPa, (c)16 GPa, and (d) 24 GPa.

∼ ∼ ∼ C11 + C22 − 2C12 > 0 ∼ ∼ ∼ C11 + C33 − 2C13 > 0 ∼ ∼ ∼ C33 + C22 − 2C23 > 0 ∼ ∼ ∼ ∼ ∼ ∼ C11 + C22 + C33 + 2C11 + 2C13 + 2C23 > 0

While the resistance to shear stress along the [1 1 0] direction became weaker when the pressure was increased from 20 GPa to 24 GPa. It is known that the bulk and shear modulus can be defined as the resistance to the volume variation and the shear stress, respectively. Generally, larger values of B and G correspond to a stronger tolerance to volume variation and shear stress, respectively. It can be observed from Fig. 2(b) that the value of B increases with increasing pressure, indicating that the resistance to volume change becomes stronger with increasing pressure. In contrast, the tolerance to shear stress becomes weaker, as evidenced by the decrease in the value of G from 20 GPa to 24 GPa. Theoretically, the stiffness of a material can be characterized by Young’s modulus. Normally, a larger value of E indicates a stiffer material. It was found that the stiffness became higher with increasing pressure, except for the case in the range from 20 GPa to 24 GPa. Furthermore, it can be observed from Fig. 2(a) and (b) that the elastic constants C11 and C66 closely correlate with the elastic modulus G and E, which tend to decrease at pressures from 20 GPa to 24 GPa. The value of Pugh's ratio B/G can be employed to evaluate the ductility and brittleness of a material. Generally, ductile materials exhibit B/G values greater than 1.75, while brittle materials exhibit values of B/G less than 1.75. It can be seen from Fig. 3 that the B/G value increases as the applied pressure was increased from 0 to 16 GPa, and the transition from brittleness to ductility occurs at 16 GPa. However, the value of B/G fell in the pressure ranges from 16 to 20 GPa and from 24 to 28 GPa, indicating that the CaMgSi compound becomes brittle at 20 GPa with a B/G value less than 1.75. Poisson's ratio v closely correlates with the compressibility of a material. Chemical bonds can be

(8)

∼ ∼ where Cii = C11 − P (i = 1, 2, 3, 4, 5, 6) ∼ ∼ ∼ ∼ ∼ ∼ C12 = C12 + P C13 = C13 + P C23 = C23 + P It is known that the values of C11, C22, and C33 reflect the resistance to linear stress along the [1 0 0], [0 1 0], and [0 0 1] directions, respectively. Normally, larger values of C11, C22, and C33 imply stronger tolerance to compression along the [1 0 0], [0 1 0], and [0 0 1] directions, respectively. It can be observed from the trend in Fig. 2(a) that the values of C22 and C33 increase with increasing external pressure, indicating that the resistance to linear stress along the [0 1 0] and [0 0 1] directions increases with the applied pressure. Nevertheless, the value of C11 decreases in the range of pressures from 20 GPa to 24 GPa, implying that the resistance to shear stress along the [1 0 0] direction at 24 GPa is weaker than that at 20 GPa. Meanwhile, it is obvious that the value of C22 is larger than that of C11 and C33, indicating that the bonding strength along [0 1 0] direction is stronger than that of along [1 0 0] and [0 0 1] directions. Moreover, the resistance to shear stress in the (1 0 0) plane along the [0 0 1] and [1 1 0] directions can be obtained by the values of C44 and C66, respectively [32]. It was found that the tolerance to shear stress along the [0 0 1] direction in the (1 0 0) plane is stronger because the value of C44 increases with increasing pressure. 5

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(a)

(b)

(c)

(d)

Fig. 7. The surface construction of bulk modulus at (a) 0 GPa, (b) 8 GPa, (c)16 GPa, and (d) 24 GPa.

AE = (EV − ER) (EV + ER)

defined as the strong interactions between the atoms or ions in a crystal. Stronger bonds correspond to greater interaction between the atoms, making it more difficult for the atoms to be separated. Generally, the stronger the chemical bonds are, the higher the hardness is. The hardness of a material with pure ionic bonds is lower than that in a comparable crystal with pure covalent bonds. However, higher pressure is beneficial for the formation of ionic bonds and leads to increased ductility, and the hardness decreases accordingly. Hence, the metallic behavior of CaMgSi becomes stronger with increasing external pressure, except for the abnormal situation at a pressure of 20 GPa. Generally, the material exhibits covalent and metallic natures at Poisson's ratio values of 0.1 and 0.33, respectively [33]. It can be observed that the nature of CaMgSi is neither metallic nor covalent at 0 GPa because the value of Poisson's ratio is 0.17 at 0 GPa. However, the metallic nature becomes stronger with increasing pressure. Meanwhile, the hardness decreases with increasing pressure except for the case of 20 GPa, which is in agreement with the above analysis.

A = AB = AG = AE = 0 indicates that the crystal is isotropic as a bulk material. Larger values of AU, AB, AG, and AE indicate higher elastic anisotropy. It can be observed from Fig. 4 that the changing tendencies of AG, AE, and AU with applied pressure are almost the same, exhibiting the highest and lowest elastic anisotropy at pressures of 24 GPa and 4 GPa, respectively. While percent anisotropy of the bulk (AB) exhibited the highest and lowest anisotropy at the pressures of 28 GPa and 4 GPa, respectively. It should also be noted that the percent anisotropy for the anisotropic index AU, bulk (AB), shear (AG) and Young’s modulus (AE) increase with increasing pressure overall except in the pressure ranges from 0 to 4 GPa and from 24 to 28 GPa. Therefore, it can be concluded that the elastic anisotropy of CaMgSi increases with increasing pressure, while an elastic anisotropic transition occurs at special pressures, such as 4 GPa and 28 GPa. This will provide beneficial guidance for further theoretical investigations and practical applications. In order to further investigate the elastic anisotropy of CaMgSi, 3D surface constructions and projections of the elastic modulus are plotted to describe the variations in elastic anisotropy in corresponding crystallographic directions. The crystallographic direction of Young's modulus E, shear modulus G, and bulk modulus B can be expressed based on the following equations [37]:

Anisotropy of elastic modulus It is important to further investigate the material’s anisotropic behavior, which has a close correlation with its physical properties [34]. Generally, the universal anisotropic index AU, percent anisotropy in bulk (AB), shear modulus (AG), and Young’s modulus (AE) can be employed to characterize the elastic anisotropy [35,36] as expressed by the following formulae:

AB = (BV − BR) (BV + BR)

(9)

AG = (GV − GR) (GV + GR)

(10)

(11)

U

1 E = l14 S11 + l 24 S22 + l34 S33 + l 22 l32 (2S23 + S44 ) + l12 l32 (2S13 + S55) + l12 l 22 (2S12 + S66)

(12)

1 B = (S11 + S12 + S13) l12 + (S12 + S22 + S23) l 22 + (S13 + S23 + S33) l32 (13) 6

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Y[100]

Y[010]

G[100]

G[010]

B[100]

B[010]

Fig. 8. The projections of Young's, shear, and bulk modulus on the (0 0 1) and (0 1 0) crystal planes of CaMgSi.

deviation of the projection in different planes from round corresponds to higher anisotropy. It can be seen that the anisotropy of Young’s modulus in the (1 0 0) and (0 1 0) planes increases with increasing pressure. For the shear modulus, the anisotropy in the (1 0 0) plane has no obvious variation. However, the anisotropy in the (0 1 0) plane exhibits no variation in the range from 0 to 8 GPa but becomes larger in the range from 8 to 24 GPa. As for the bulk modulus, the anisotropy shows no significant change in the (1 0 0) plane, while it becomes smaller with increasing pressure.

1 G = 2S11 l12 (1 − l12) + 2S 22l 22 (1 − l 22) + 2S33 l32 (1 − l32) − 4S12 l12 l 22 −

4S13 l12 l32 1

1

− 4S23 l 22 l32 + 2 S44 (1 − l12 − 4l 22 l32) + 2 S55 (1 − l 22 − 4l12 l32) 1

+ 2 (1 − l32 − 4l12 l 22) (14) Generally, larger deviations from the sphere correspond to a greater degree of anisotropy. It can be observed from Figs. 5–7 that the elastic anisotropy increases with increasing pressure overall. More details of the elastic anisotropy in different planes can be described by the projection of the elastic modulus, as shown in Fig. 8. Normally, a larger 7

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(b)

(a)

(c)

(d)

Fig. 9. PDOS for CaMgSi (a) Ca, (b) Mg, and (c) Si. (d) TDOS of CaMgSi at 0, 8, 16 and 24 GPa.

addition, it can be observed from Fig. 9 that the valence band structure changes considerably in the range from 0 to 24 GPa. With increasing pressure, the peak value of the TDOS becomes smaller with increasing pressure, the edge of the valence band moves to a lower energy region, and the conduction band shifts to a higher energy region. The following observations can be made about the properties of CaMgSi at higher pressures: (1) the density of states becomes broader for Ca, Mg, and Si. (2) The valence and conduction bands shift to lower and higher energy, respectively, for Ca, Mg, and Si atoms. (3) The valence peak value of the density of states near the Fermi level remains almost constant and the conduction peak value near the Fermi level becomes lower for Ca atoms. (4) For Mg atoms, the valence peak value of the density of states near the Fermi level decreases for pressures ranging from 0 to 8 GPa,

Electronic thermoelectric properties To further understand the structural stability of CaMgSi, the partial density of states (PDOS) per atom near the Fermi level and the total density of states (TDOS) were calculated to determine the bonding interactions under different pressures, as shown in Fig. 9. Based on the TDOS at the Fermi level, the values of the pseudo-gap under pressures of 0, 8, 16, and 24 GPa are 3.229, 3.435, 3.717, and 4.326 eV, respectively. Generally, a larger pseudo-gap at the Fermi level indicates a structure with higher stability. Therefore, it can be confirmed that the structural stability of crystalline CaMgSi increases with increasing pressure. Meanwhile, CaMgSi exhibits stronger metallic behavior due to the increasing value of the TDOS at zero with increasing pressure. In 8

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(a)

(b)

(d)

(c)

Fig. 10. Charge density difference of CaMgSi at (a) 0 GPa, (b) 8 GPa, (c) 16 GPa, and (d) 24 GPa for the (0 1 0) plane.

be obtained based on the following formula:

rises in the range of 8–16 GPa, and then decreases again at 16–24 GPa, and the conduction peak value near the Fermi level decreases with pressure. (5) The valence peak value of the density states near the Fermi level becomes lower while the conduction peak value near the Fermi level almost does not change, which can be attributed the variation of the crystal structure with the increasing applied pressure. Moreover, the orbitals that are not hybridized can become hybridized with increased pressure, leading to stronger structural stability. The difference in charge density along the (0 1 0) plane is plotted in Fig. 10 to describe the bonding characteristics of CaMgSi. Theoretically, the covalent and ionic bonds between two atoms can be predicted by the negative and positive charges at the atomic positions [38]. A scale bar is shown at the right side (color line) of the contour plot in units of e/Å3. The high and low electron densities correspond to the colors red and blue, respectively. Obviously, the charge density is mainly distributed around the Si and Ca atoms, indicating the ionic nature of Ca-Si bonds. Meanwhile, it can be concluded that the CaMgSi compound exhibits metallic characteristics due to the large amount of Ca-Si ionic bonds distributed in the CaMgSi compound. Bonds of a covalent nature can be observed between Mg and Si atoms due to the hybridization found between Mg and Si atoms, as shown in Fig. 9. Meanwhile, the charge density difference along different planes has also been investigated and the results support the present conclusions. Moreover, it can be observed that the charge density between Ca and Si becomes larger with increasing pressure, indicating that a large number of Ca-Si covalent bonds are formed with increasing pressure, leading to stronger structural stability, which is well consistent with the analysis of the density of states presented above. The value of ZT (the thermoelectric figure of merit) can be employed to evaluate the performance of a thermoelectric material. It can

ZT = Sσ 2T K

(15)

where S represents the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, and K is the thermal conductivity, which consists of contributions from the electronic and lattice thermal conductivities, ke and kl, respectively. As is well known, the density of energy states reflects the number of states occupied by electrons within a certain energy range. Based on the theory of electronic transport, the conductivity is only attributed to the electrons near the Fermi surface, and a larger energy state density near the Fermi surface indicates a higher number of states occupied by electrons and a higher carrier density, corresponding to a higher electrical conductivity σ and electronic thermal conductivity ke accordingly. Hence, it is important to deeply investigate the TDOS and DOS of CaMgSi in terms of its thermoelectric properties. It can be observed from Fig. 9(d) that the peak value of the total density of states near the Fermi surface (−2 ∼ 2 eV) decreases with increasing pressure. Moreover, the variation of the peak value of the density of states from 0 to 8 GPa is considerable, but only moderate from 8 to 24 GPa. Generally, a lower peak value of the DOS near the Fermi surface (−2 ∼ 2 eV) indicates reduced hopping of carriers and that fewer electrons contributing to the electrical conductivity can be accommodated, corresponding to a lower value of the power factor and lower electrical conductivity. Therefore, the electrical conductivity and electronic thermal conductivity decrease with increasing pressure. Moreover, the total density of states is mainly composed of contributions from the 3d orbital electrons of Ca, 3p orbital electrons of Mg, and 3p orbital electrons of Si, as shown in Fig. 9(d). Meanwhile, the lattice thermal conductivity kl can be obtained 9

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(b) (a)

(c)

Fig. 11. (a) Calculated Debye temperature ΘD(K) at 0 K and average vm (m/s), (b) the Grüneisen parameter γɑ, and (c) phonon density of states.

phonon density of states at pressures of 0, 8, 16, and 24 GPa is shown in Fig. 11(c). It can be observed that the phonon DOS is transferred to the higher frequency region, which also implies that the thermal stability becomes stronger with higher pressure overall. The Grüneisen parameter γɑ can describe the anharmonic correlation between atoms in solids [41]. Normally, a larger value of γɑ implies a stronger dependence of the crystal volume on the temperature (or pressure). As is shown in Fig. 11(b), the calculated values of γɑ increase with increasing pressure, but decrease in the ranges of 16–20 GPa and 24–28 GPa, indicating that the effect of temperature (or pressure) on the variation of volume is weaker in these two regions. The enthalpy E(T), Gibbs free energy G(T), and entropy S(T) of the CaMgSi compound under pressures from 0 to 24 GPa are shown in Fig. 12(a–c). The values of enthalpy E(T) and entropy S(T) increase with increasing temperature and decrease with increasing pressure, while the Gibbs free energy G(T) decreases with increasing temperature and increases with increasing pressure. The variations of G(T) and S(T) under different pressures indicate that the most considerable effect of pressure on the Gibbs free energy G(T) and entropy S(T) occurs between 8 and 16 GPa. In addition, the smallest change in the enthalpy E(T) was observed between 0 and 8 GPa, implying that the effect of pressure on enthalpy E(T) is weak at low pressure. Moreover, Gibbs free energy G(T) can be a criterion to measure the thermal stability of materials [42,43]. Generally, a lower value of Gibbs free energy corresponds to better thermal stability. It can be seen from Fig. 12(b) that the Gibbs free energy increases with increasing pressure, indicating that the thermal stability of a CaMgSi crystal becomes weaker with increasing pressure. The value of the heat capacity can be employed to evaluate the vibrational properties of bulk materials. As is shown in Fig. 12(c), the volume heat capacity decreases with increasing pressure at constant temperature. Theoretically, the volume heat capacity is proportional to T3 below the Debye temperature. And the value of the constant volume

based on the following formula:

kl = c v vph lph

(16)

where Cv, vph, and lph represent the heat capacity, phonon velocity, and mean free path of a phonon, respectively. Hypothesizing the invariance of vph and lph, a lower heat capacity corresponds to a lower lattice thermal conductivity, which is beneficial for the thermoelectric property. Based on the PDOS, the calculated values of Cv are shown in Fig. 11(d). It can be seen that the heat capacity decreases with increasing pressure, especially at pressures from 8 to 16 GPa, indicating that lower values of the lattice thermal conductivity kl result in the improvement of the thermoelectric figure of merit ZT. Debye temperatures and thermodynamic properties It is known that thermoelectric devices are operated at high temperatures. Hence, it is essential to further investigate the thermodynamic properties of thermoelectric materials at various temperatures and pressures ranging from 0 to 24 GPa, such as enthalpy E(T), Gibbs free energy G(T), heat capacity CV(T), entropy S(T), Grüneisen parameter, and Debye temperature ΘD(K), which can give comprehensive guidance for theoretical work and the practical application of CaMgSi. However, first-principles calculations based on DFT are limited to a temperature of absolute zero, T = 0 K, but basic thermodynamic properties can be obtained from phonon calculations. The Debye temperature is related to the binding forces between neighboring atoms [39]. Generally, a larger value of the Debye temperature ΘD corresponds to stronger covalent bonds [40]. As is shown in Fig. 11(a), the values of the Debye temperature increase with increasing pressure, except at pressures from 20 to 24 GPa, indicating that the binding force becomes stronger with increasing pressure, but weaker at the special pressure range from 20 to 24 GPa. Meanwhile, the 10

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(b)

(a)

(d)

(c)

Fig. 12. (a) Calculated enthalpy E(T), (b) Gibbs free energy G(T), (c) heat capacity at constant volume CV(T), and (d) entropy S(T) of the CaMgSi compound under pressures from 0 to 24 GPa.

CaMgSi decrease with pressure, according to the calculation of the TDOS. On one hand, the reduction of electrical conductivity is bad for thermoelectric performance. On the other hand, the decrease in the electronic thermal conductivity is beneficial for thermoelectric performance. In addition, the value of the heat capacity decreases with increasing pressure, indicating that the reduction of the lattice thermal conductivity is beneficial for thermoelectric performance, especially at pressures from 8 to 16 GPa. (6) The values of the Debye temperature, enthalpy, and entropy increase with increasing pressure, indicating that the structural stability can be enhanced at higher pressure. However, the thermal stability of CaMgSi becomes weaker due to the larger value of the Gibbs free energy at higher pressures.

heat capacity converges to the Dulong–Petit limit of 3NR when the temperature is higher than the Debye temperature. The heat capacity of CaMgSi converges to values of 68.91, 67.42, 65.20, and 64.06 J mol−1 K−1 at pressures of 0, 8, 16, and 24 GPa, respectively. Conclusion In summary, the structure, electronic, elastic, elastic anisotropy, thermoelectric, and thermodynamic properties of CaMgSi under different pressures were obtained by first-principles methods. The optimized structure and lattice parameters are well consistent with previous experimental results, indicating that the calculations in this work are feasible and reliable. The conclusions can be summarized as follows:

Acknowledgements

(1) The values of the bulk modulus B, Young’s modulus E, and shear modulus G increase with increasing pressure, whereas the values of G and Y decrease in the range from 20 to 24 GPa. (2) The values of v and B/G increase with increasing pressure, implying that CaMgSi is prone to change from brittle to ductile with increasing pressure. The transition from brittleness to ductility occurs at a pressure of 16 GPa. The values of Hv decrease with increasing pressure, except for the ranges of 16–20 GPa and 24–28 GPa. (3) Universal elastic anisotropy increases at pressures from 4 to 24 GPa, but decreases at pressures from 0 to 4 and from 24 to 28 GPa. Moreover, the variation in the elastic anisotropy for G and E is larger than that of B. (4) The bonding behaviors of the CaMgSi compound consist of a combination of Mg-Si covalent bonds and Ca-Si ionic bonds at 0 GPa. Furthermore, Ca-Si covalent bonds are formed with increasing pressure. (5) The electrical conductivity and electronic thermal conductivity of

The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Nos. 51665036, 51264032), the Natural Science Foundation of Jiangxi Province (No. 20152ACB20014). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.rinp.2019.102483. References [1] Chen Z, Jian Z, Li W, Chang Y, Ge B, Hanus R, et al. Lattice dislocations enhancing thermoelectric PbTe in addition to band convergence. Adv Mater 2017;29:1606768. [2] Wan L, Beckman S. First-principles investigations on the thermoelectric properties of Bi2Te3 doped with Se. Mater Res Soc Symp Proc 2013;1543:23–8.

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