- Email: [email protected]
First-principles calculations of the structural and elastic properties of β-FeSi2 at high-pressure
Intermetallics 18 (2010) 1222e1227
Contents lists available at ScienceDirect
Intermetallics journal homepage: www.elsevier.com/locate/intermet
First-principles calculations of the structural and elastic properties of b-FeSi2 at high-pressure Jun-ichi Tani*, Masanari Takahashi, Hiroyasu Kido Electronic Materials Research Division, Osaka Municipal Technical Research Institute, 1-6-50 Morinomiya, Joto-ku, Osaka 536-8553, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 September 2009 Received in revised form 16 February 2010 Accepted 11 March 2010 Available online 3 April 2010
The pressure dependence of the structural and elastic properties of b-FeSi2 in the range 0e60 GPa was investigated using first-principles calculations based on density functional theory. Calculations were performed within the local density approximation as well as the generalized gradient approximation to the exchange correlation potential. The calculated lattice constants and internal parameters are in good agreement with previous experimental results. The nine independent elastic constants, c11, c22, c33, c44, c55, c66, c12, c13, and c23, of orthorhombic b-FeSi2 have been evaluated. The isotropic bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, elastic anisotropy, and Debye temperature of polycrystalline b-FeSi2 under pressure are also presented. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: A. Silicide, various B. Elastic properties B. Anisotropy E. Ab-initio calculations E. Phase stability, prediction
1. Introduction Transition metal silicides are of considerable interest from the viewpoint of their structural and functional applications [1]. The semiconducting phase of iron disilicide (b-FeSi2) has been studied as a material for thermoelectric conversion, solar cells, and optoelectronic applications [2e5]. Dusausoy et al. [6] reported the crystal structure of b-FeSi2 to be a base-centered orthorhombic system (space group: Cmca) having 48 atoms per unit cell with lattice constants a ¼ 0.9863 nm, b ¼ 0.7791 nm, and c ¼ 0.7833 nm Fig. 1 shows the unit cell of b-FeSi2. There are two crystallographically inequivalent sites for Fe and Si (FeI, FeII, SiI, and SiII). The unit cell contains 16 formula units distributed over 8 FeI, 8 FeII, 16 SiI, and 16 SiII. The two types of Fe sites are coordinated by 8 Si atoms with slightly different distances to Fe, and have the point symmetries 2 (C2) and m (Cs), respectively. Takarabe et al. [7e9] and Mori et al. [10] reported the pressure dependence of the structural and optical properties of b-FeSi2. They [9,10] pointed out that though a-FeSi2 is denser than b-FeSi2, its bulk modulus is only 75% that of b-FeSi2. Moreover, high-pressure X-ray diffraction measurements revealed that a high-pressure phase starts to appear at about 20 GPa. The elastic properties are very important to understanding solid-state physical, chemical, and mechanical
* Corresponding author. Tel.: þ81 6 6963 8081; fax: þ81 6 6963 8099. E-mail address: [email protected] (J.-i. Tani). 0966-9795/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2010.03.023
properties. Although the bulk modulus of b-FeSi2 has been investigated via theory and experiment, to the best of our knowledge, no other elastic constant of b-FeSi2 have been reported. Moreover, it is important to investigate the structural and elastic properties of b-FeSi2 at high-pressure so as to understand its phase stability. In the present study, to investigate b-FeSi2 on the first-principles, we performed calculations using the density functional pseudopotential method to obtain precise information on the structural and elastic properties of this system in the pressure range of 0e60 GPa. 2. Computational details Density functional theory (DFT) calculations were performed using the computer program CASTEP (Cambridge serial total energy package in material modeling, Accelrys Inc.) [11]. Here, we used the norm-conserving potential generated by means of the optimization scheme of Lin et al. [12]. The iron 3d64s2 orbitals and silicon 3s23p2 orbitals were treated as valence states. Two approximations for the exchange correlation potential were utilized, namely, the local density approximation (LDA) and the generalized gradient approximation (GGA). For the exchange correlation potentials, we used the CeperleyeAlder form of the LDA as parameterized by Perdew and Zunger [13], and for the GGA form we used the Perdew and Wang [14] interpolation formula, known as PW91. We constructed a base-centered orthorhombic primitive cell having 24 atoms (4 FeI, 4 FeII, 8 SiI, and 8 SiII). We expanded the
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
1223
Table 1 Structural properties of the calculated b-FeSi2, obtained from LDA as well as GGA calculations, in comparison with previous experimental data [6]. Deviations from the experimental data are shown in parentheses. Structural parameters
Calculated LDA
Cell parameters
a (Å) b (Å) c (Å) V (Å3) Atomic coordinates FeI (8d)
Fig. 1. Unit cell of b-FeSi2. The large spheres represent Fe and the small spheres represent Si. There are two crystallographically inequivalent sites of both Fe and Si (FeI, FeII, SiI, SiII).
valence electronic wave functions in a plane wave basis set up to an energy cutoff of 560 eV, which converges the total energy of the unit cell to better than 1 meV/atom. The density mixing method was used for electronic minimization. In the total energy calculations, integrations over the Brillouin zone were performed by using a 4 4 3 MonkhorstePack set [15], which produced 12 irreducible k points in the Brillouin zone of the 24-atom primitive cell. The structure was examined as a function of external hydrostatic pressure ranging from 0 to þ60 GPa. The elastic stiffness coefficients were determined using a finite strain technique [16] and a linear fit of the computed stress for small strains, where 6 strain amplitudes up to a maximum of 0.3% were used. 3. Results and discussion Table 1 compares the structural properties calculated for b-FeSi2 by the LDA and GGA calculations with previous experimental data [6]. The LDA results for lattice constants a, b, and c are 0.9%, 1.0%, and 0.9% smaller than the previous experimental results, which are within the range of typical LDA errors. The GGA results for lattice constants a, b, and c are in excellent agreement (within 0.2%) with the previous experimental results. In both LDA and GGA, the calculated internal parameters (x, y, z) of FeI, FeII, SiI, and SiII atoms are also in excellent agreement with the previous experimental results. Fig. 2 shows the pressure dependence of the normalized lattice parameters a/a0, b/b0, and c/c0 (a) as well as the normalized unit cell volume V/V0 (b) (where a0, b0, c0, and V0 are the equilibrium structural parameters at pressure P ¼ 0 GPa). The normalized
9.7734 7.7146 7.7663 585.56 0.2156 0 0 FeII (8f) 0.5 0.3085 0.1856 SiI (16g) 0.1285 0.2742 0.0516 SiII (16g) 0.3734 0.0456 0.2265
Experimental [6] GGA
(0.9%) (1.0%) (0.9%) (2.7%)
9.8692 7.7741 7.8455 601.94 0.2160 0 0 0.5 0.3082 0.1849 0.1282 0.2744 0.0516 0.3731 0.0457 0.2266
(þ0.1%) 9.863 (0.2%) 7.791 (þ0.2%) 7.833 (0.0%) 601.91 0.2146 0 0 0.5 0.3086 0.1851 0.1282 0.2746 0.0512 0.3727 0.0450 0.2261
volume V/V0 decreases with increasing pressure. The V/V0 values for b-FeSi2 at P ¼ 60 GPa from our LDA and GGA calculations are 0.814 and 0.802, respectively. The ratio of decline in LDA results is smaller than that in GGA results. The a/a0, b/b0, and c/c0 values for b-FeSi2 at P ¼ 60 GPa from our LDA (GGA) calculations are 0.923 (0.917), 0.943 (0.939), and 0.936 (0.931), respectively. We note that the order of compressibility is a-axis > c-axis > b-axis at P ¼ 0e60 GPa. Takarabe et al. [9] pointed out that the normalized linear compressibility of the a-axis is about 20%e30% greater than those of the c- and b-axes, indicating that the bonding along the a-axis is weaker than those along the other two axes. In order to understand the bonding, we calculated the nearest-neighbor distances of FeIeSiI, FeIeSiII, FeIIeSiI, FeIIeSiII, SiIeSiI, SiIeSiII, SiIIeSiII, and FeIeFeII of relaxed b-FeSi2 (Table 2). At P ¼ 60 GPa, the normalized nearest-neighbor distances of FeIeSiI, FeIeSiII, FeIIeSiI, FeIIeSiII, SiIeSiI, SiIeSiII, SiIIeSiII, and FeIeFeII are respectively 94.1% (93.6%), 94.0% (93.2%), 94.1% (93.6%), 94.3% (93.7%), 93.3% (92.8%), 92.9% (92.5%), 92.5% (92.0%), and 90.5% (90.1%) with respect to those at P ¼ 0 GPa. Therefore, the order of compressibility is FeeFe > SieSi > FeeSi. The directions of the FeeFe, SieSi, and FeeSi bonds in b-FeSi2 greatly affect the compressibility of the a-, b-, and c-axes. The internal parameters (x, y, z) for FeI, FeII, SiI, and SiII atoms of b-FeSi2 at P ¼ 0 GPa from our LDA calculations are respectively (0.2156, 0, 0), (0.5, 0.3085, 0.1856), (0.1285, 0.2742, 0.0516), and (0.3734, 0.0456, 0.2265), while those at P ¼ 60 GPa are respectively (0.2101, 0, 0), (0.5, 0.3141, 0.1796), (0.1308, 0.2767, 0.0594), and (0.3735, 0.0510, 0.2209). The internal parameters for FeI, FeII, SiI, and SiII are slightly affected by the pressure. Therefore, the bond bending in b-FeSi2 might slightly affect the compressibility of the a-, b-, and c-axes. Table 3 shows the pressure dependence of elastic stiffness (cij) of b-FeSi2 obtained from LDA and GGA calculations. The elastic stiffness tensor cij of orthorhombic crystals has nine independent components, c11, c22, c33, c44, c55, c66, c12, c13, and c23. For orthorhombic crystals, mechanical stability requires the elastic constants satisfy the well-known Born stability criteria [17e19]: cii P > 0 (i ¼ 1e6), (c11 þ c22 2c12 4P) > 0, (c11 þ c33 2c13 4P) > 0, (c22 þ c33 2c23 4P) > 0, (c11 þ c22 þ c33 þ 2c12 þ 2c13 þ 2c23 9P) > 0
(1)
The elastic constants in Table 3 satisfy all of these conditions, which suggests that all the calculated structures in the range of 0e60 GPa are elastically stable. Because no experimental or
1224
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227 Table 2 Mean nearest-neighbor distances of FeeSi, SieSi, and FeeFe of b-FeSi2 at P ¼ 0 and 60 GPa, obtained from LDA as well as GGA calculations. The value in parentheses represents the percentage of the distances at P ¼ 60 GPa with respect to those at P ¼ 0 GPa. Method
LDA
Pressure (GPa) Mean nearest-neighbor FeIeSiI FeIeSiII distance (Å) FeIIeSiI FeIIeSiII SiIeSiI SiIeSiII SiIIeSiII FeIeFeII
0 2.3318 2.3439 2.3651 2.3541 2.5230 2.5102 2.4571 2.9498
GV ¼
GGA 60 2.1948 2.2032 2.2257 2.2190 2.3536 2.3331 2.2738 2.6681
(94.1%) (94.0%) (94.1%) (94.3%) (93.3%) (92.9%) (92.5%) (90.5%)
0 2.3525 2.3656 2.3870 2.3764 2.5478 2.5323 2.4811 2.9784
60 2.2030 2.2042 2.2347 2.2277 2.3636 2.3427 2.2831 2.6831
(93.6%) (93.2%) (93.6%) (93.7%) (92.8%) (92.5%) (92.0%) (90.1%)
1 1 ðc þ c22 þ c33 c12 c13 c23 Þ þ ðc44 þ c55 þ c66 Þ 15 11 5 (3)
Reuss’s bulk modulus (BR) and Voigt’s bulk modulus (BV) of the orthorhombic system are given by equations (4) and (5):
BR ¼
1 ðs11 þ s22 þ s33 Þ þ 2ðs12 þ s13 þ s23 Þ
(4)
BV ¼
1 2 ðc þ c22 þ c33 Þ þ ðc12 þ c13 þ c23 Þ 9 11 9
(5)
In equations (2) and (4), the sij are the elastic compliance, which is converted from cij using the relation between cij and sij in the orthorhombic system. Hill [22] has proved that the Voigt and Reuss equations represent the upper and lower limits of the polycrystalline constants, and the arithmetic mean value of the Voigt and Reuss moduli, GH ¼ (GR þ GV)/2 and BH ¼ (BRþBV)/2, are taken to estimate the shear and bulk moduli. E and n can be calculated using Hill’s shear and the bulk moduli (GH and BH), and they are given by equations (6) and (7):
E ¼
9BH GH 3BH þ GH
(6)
n¼
3BH 2GH 2ð3BH þ GH Þ
(7)
The values of B, G, E, and n are shown in Table 4. The BH value for b-FeSi2 at P ¼ 0 GPa from our LDA (GGA) calculations is 194.5 Fig. 2. Normalized lattice constants a/a0, b/b0, and c/c0 (a) and normalized unit cell volume V/V0 (b) of b-FeSi2 as a function of pressure P. The results from LDA and GGA calculations are represented by the solid and dashed curves, respectively.
theoretical data for the cij of b-FeSi2 are available in previous reports, the present work is the first theoretical prediction of the value of cij. From our calculated cij results, the isotropic shear modulus G, bulk modulus B, Young’s modulus E, and Poisson’s ratio n of polycrystalline b-FeSi2 were determined using the VoigteReusseHill averaging scheme [20e22]. According to the Voigt and Reuss approximation, Reuss’s shear modulus (GR) and the Voigt’s shear modulus (GV) in the orthorhombic system are given by equations (2) and (3) [23]:
GR ¼
15 4ðs11 þ s22 þ s33 Þ 4ðs12 þ s13 þ s23 Þ þ 3ðs44 þ s55 þ s66 Þ (2)
(172.8) GPa, which is in good agreement with previously calculated values (199 GPa by Christensen [24], 203 GPa by van Ek et al. [25], and 182e209 GPa by Moroni et al. [26]), and is w20%e29% lower than a previous experimental value determined from high-pressure X-ray diffraction measurements (243.5 GPa) [9,10]. The calculated BH value for b-FeSi2 increases with increasing pressure, and at the pressure of 60 GPa, the BH value from our LDA (GGA) calculations is 416.8 (399.4) GPa. The BH values at 60 GPa are 2.1e2.3 times larger than those at P ¼ 0 GPa. Pugh [27] reported that the quotient of bulk to shear modulus (B/G) is an indication of the extent of the plastic range of a pure metal, so that a high B/G value is associated with ductility and a low B/G value with brittleness. The critical value that separates ductile and brittle materials is w1.75. The calculated BH/GH values at P ¼ 0 GPa for b-FeSi2 from our LDA (GGA) calculations is 1.39 (1.33), suggesting that b-FeSi2 is brittle. However, the calculated BH/GH values increase with increasing pressure and the value at 60 GPa from our LDA (GGA) calculations is 1.83 (1.86), suggesting that b-FeSi2 becomes more ductile with increasing pressure.
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
1225
Table 3 Pressure dependence of elastic stiffness (cij) of b-FeSi2 obtained from LDA and GGA calculations. P (GPa) 0
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
10 20 30 40 50 60
c11 (GPa)
c22 (GPa)
c33 (GPa)
c44 (GPa)
c55 (GPa)
c66 (GPa)
c12 (GPa)
c13 (GPa)
c23 (GPa)
360.8 314.6 418.1 376.3 478.8 446.0 532.3 490.4 578.2 533.6 637.1 590.3 672.6 640.5
393.9 355.8 473.6 424.7 535.3 491.6 597.8 551.8 655.6 611.6 710.9 644.2 767.3 721.8
397.2 362.5 477.1 419.6 526.0 482.3 584.8 545.3 643.0 603.6 697.2 658.5 746.7 705.7
144.8 126.4 165.2 148.3 182.5 168.5 201.6 183.1 219.9 201.9 233.8 218.3 250.3 232.8
131.8 127.0 154.2 141.9 170.7 157.2 181.6 169.7 194.0 181.8 202.1 190.8 214.5 205.5
142.8 142.0 164.8 152.7 172.5 164.5 184.2 178.3 194.4 192.2 207.5 196.6 214.2 206.6
102.0 95.8 130.8 126.1 156.8 156.4 184.8 182.3 211.1 206.7 238.7 234.3 264.9 261.3
85.1 77.7 114.8 102.4 137.3 132.4 162.5 155.7 185.8 182.5 213.6 206.7 233.8 230.5
114.2 89.4 143.0 126.2 173.6 158.5 204.2 188.3 234.1 219.8 261.4 247.8 290.2 276.2
The shear anisotropic factors are given by [23]
A1 ¼
4c44 c11 þ c33 2c13
for the f100g planes
(8)
A2 ¼
4c55 c22 þ c33 2c23
for the f010g planes
(9)
A3 ¼
4c66 c11 þ c22 2c12
for the f001g planes
(10)
The calculated values of A1, A2, and A3 for b-FeSi2 are given in Table 5. A value of 1 means completely isotropic properties, whereas values smaller or greater than 1 measure degrees of anisotropy. It can be seen that b-FeSi2 exhibits low anisotropy at P ¼ 0e60 GPa, because the values of A1, A2, and A3 are in the range of 0.91e1.12, which is close to 1. Using the Voigt and Reuss moduli, the percentage anisotropies in the compressibility and shear (AB and AG) are defined as [28]
AB ¼
BV BR 100 BV þ BR
(11)
AG ¼
GV GR 100 GV þ GR
(12)
A value of 0% implies completely isotropic properties, whereas 100% represents the maximum anisotropy. The calculated values of AB and AG for b-FeSi2 are given in Table 5. It can be seen that the
anisotropies in both compressibility and shear are also small, because the values of AB and AG are in the range of 0.09%e0.34%. The anisotropies of the linear bulk moduli along the a-, b-, and caxes are defined as [23]
dP L Ba ¼ a ¼ da 1þaþb
(13)
dP Ba ¼ a db
(14)
dP Ba Bc ¼ c ¼ b dc
(15)
Bb ¼ b
where
L ¼ c11 þ 2c12 a þ c22 a2 þ 2c13 b þ c33 b þ 2c23 ab 2
(16)
a ¼
ðc11 c12 Þðc33 c13 Þ ðc23 c13 Þðc11 c13 Þ ðc33 c13 Þðc22 c12 Þ ðc13 c23 Þðc12 c23 Þ
(17)
b¼
ðc22 c12 Þðc11 c13 Þ ðc11 c12 Þðc23 c12 Þ ðc22 c12 Þðc33 c13 Þ ðc12 c23 Þðc13 c23 Þ
(18)
The anisotropies of the bulk modulus along the a- and c-axes with respect to the b-axis can be written as [23]
ABa ¼
Ba ¼ a Bb
(19)
Table 4 Pressure dependence of isotropic shear modulus (G) and bulk modulus (B) of polycrystalline b-FeSi2 obtained from the elastic constants cij using the VoigteReusseHill averaging scheme. Young’s modulus (E) and Poisson’s ratio (n) are determined using the Hill’s shear and bulk moduli (GH and BH). P (GPa) 0 10 20 30 40 50 60
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
GR (GPa)
GV (GPa)
GH (GPa)
BR (GPa)
BV (GPa)
BH (GPa)
E (GPa)
n
BH/GH
140.2 129.8 161.8 145.9 176.3 162.6 190.5 176.5 203.9 190.6 216.6 200.6 227.8 214.9
140.6 130.4 162.2 146.3 176.6 162.9 191.0 177.0 204.7 191.2 217.4 201.4 229.0 215.6
140.4 130.1 162.0 146.1 176.5 162.7 190.8 176.8 204.3 190.9 217.0 201.0 228.4 215.3
194.1 172.5 237.0 213.4 273.6 256.3 311.4 291.9 346.5 327.5 383.9 361.9 415.4 398.3
194.9 173.2 238.4 214.4 275.1 257.2 313.1 293.3 348.8 329.6 385.8 363.4 418.3 400.4
194.5 172.8 237.7 213.9 274.3 256.7 312.2 292.6 347.6 328.6 384.9 362.7 416.8 399.4
339.5 312.0 396.0 357.0 435.9 403.0 475.5 441.4 512.6 479.7 548.0 509.0 579.4 547.4
0.209 0.199 0.222 0.222 0.235 0.238 0.246 0.249 0.254 0.257 0.263 0.266 0.268 0.272
1.39 1.33 1.47 1.47 1.56 1.58 1.64 1.66 1.71 1.73 1.78 1.81 1.83 1.86
1226
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
Table 5 Pressure dependence of shear anisotropic factors A1, A2, and A3; anisotropy in compressibility and shear AB and AG, respectively; linear bulk moduli Ba, Bb, and Bc along the a-, b-, and c-axes of the orthorhombic cell, respectively; and anisotropies of the bulk moduli ABa and ABc along the a- and c-axes with respect to the b-axis, respectively. P (GPa) 0
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
10 20 30 40 50 60
ABc ¼
A1
A2
A3
AB (%)
AG (%)
Ba (GPa)
Bb (GPa)
Bc(GPa)
ABa
ABc
0.985 0.969 0.993 1.004 1.000 1.016 1.018 1.011 1.035 1.046 1.031 1.045 1.052 1.052
0.937 0.942 0.928 0.959 0.956 0.957 0.938 0.942 0.934 0.938 0.913 0.946 0.919 0.939
1.037 1.186 1.046 1.113 0.985 1.053 0.969 1.053 0.958 1.051 0.953 1.027 0.941 0.984
0.22 0.21 0.30 0.25 0.26 0.18 0.28 0.25 0.32 0.33 0.25 0.21 0.34 0.27
0.13 0.24 0.12 0.14 0.09 0.09 0.14 0.12 0.19 0.15 0.20 0.19 0.25 0.18
515.1 457.6 612.1 561.7 715.5 688.3 808.8 760.7 886.3 833.0 997.6 945.0 1054.9 1027.4
642.5 569.6 793.5 730.3 929.7 866.3 1069.1 1002.2 1203.1 1137.0 1315.0 1202.8 1466.3 1388.2
604.5 538.4 754.6 650.7 846.5 772.1 961.7 897.7 1079.2 1027.2 1187.5 1144.7 1286.4 1223.7
0.802 0.803 0.771 0.769 0.770 0.795 0.756 0.759 0.737 0.733 0.759 0.786 0.719 0.740
0.941 0.945 0.951 0.891 0.911 0.891 0.900 0.896 0.897 0.903 0.903 0.952 0.877 0.882
a Bc ¼ b Bb
(20)
The calculated values of Ba, Bb, Bc, ABa, and ABc for b-FeSi2 are given in Table 5. At P ¼ 0 GPa, the directional bulk moduli Ba, Bb, and Bc from our LDA (GGA) calculations are 515.1 (457.6), 642.5 (569.6), and 604.5 (538.4) GPa, respectively. Therefore, the bulk modulus is the greatest along b-axis and the smallest along a-axis. As discussed earlier, the order of compressibility is a-axis > c-axis > b-axis. Therefore, the order of compressibility corresponds with the order of directional bulk moduli Ba < Bc < Bb. The directional bulk moduli, Ba, Bb, and Bc, increase with pressure. From the cij, we can calculate the elastic Debye temperature (QD). At T ¼ 0 K, QD equals the Debye temperature determined from specific heat measurements [29]. QD is proportional to the averaged sound velocity (vm) and can be calculated by the following equation [30]:
QD
h 3n NA r 1=3 ¼ vm k 4p M
(21)
where h is Planck’s constant, k Boltzmann’s constant, NA Avogadro’s number, r the density, M the molecular weight, and n the number of atoms per formula unit. Table 6 Pressure dependence of density (r), transverse (vt), longitudinal (vl), and average elastic velocities (vm), and Debye temperature (QD) calculated from elastic constants. P (GPa) 0 10 20 30 40 50 60
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
r (g/cm3)
vt (m/s)
vl (m/s)
vm (m/s)
QD (K)
5.082 4.944 5.335 5.211 5.547 5.443 5.742 5.648 5.921 5.835 6.087 6.006 6.242 6.165
5256 5130 5510 5295 5640 5468 5764 5594 5874 5720 5971 5785 6049 5909
8667 8369 9222 8856 9585 9329 9933 9672 10 233 9997 10 525 10 247 10 750 10 552
5808 5663 6098 5860 6251 6062 6396 6210 6525 6355 6639 6435 6730 6577
751.2 725.6 801.5 764.2 832.4 802.1 861.6 831.9 888.0 860.6 911.8 879.9 932.1 907.2
The average wave velocity (vm) in the polycrystalline material can be calculated using
vm
!#1=3 " 1 2 1 ¼ þ 3 v3t v3 l
(22)
where vt and vl are the traverse and longitudinal elastic wave velocities of the polycrystalline material, respectively. The values of the average traverse and longitudinal sound velocities (vt and vl) can be obtained using Hill’s shear modulus (GH) and the bulk modulus (BH) as follows [31]:
vt ¼
sffiffiffiffiffiffiffi GH
r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 r BH þ GH vl ¼ 3
(23)
(24)
The calculated values of r, vt, vl, vm, and QD for b-FeSi2 are given in Table 6. At P ¼ 0 GPa, QD of b-FeSi2 from our LDA (GGA) calculations is 752.1 (725.6) K. The value is w15%e20% higher than the previous experimental value of 630 K obtained from analysis of specific heat by Waldecker et al. [32]. The Debye temperature of bFeSi2 increases monotonically with increasing pressure. 4. Conclusions The pressure dependence of the structural and elastic properties of b-FeSi2 in the range 0e60 GPa was investigated using firstprinciples calculations based on density functional theory. The calculated elastic constants indicate that b-FeSi2 is elastically stable in the range of 0e60 GPa. The BH value for b-FeSi2 at P ¼ 0 GPa from our LDA (GGA) calculations is 194.5 (172.8) GPa, which is in good agreement with previously calculated values, and is w20%e29% lower than a previous experimental value determined from highpressure X-ray diffraction measurements (243.5 GPa). Further experimental studies will be necessary in order to settle this discrepancy between theoretical calculations and experimental results. The bulk modulus of b-FeSi2 is the greatest along b-axis and the smallest along a-axis. However, we predict that anisotropies in both compressibility and shear are small. The calculated BH/GH values at P ¼ 0 GPa for b-FeSi2 from our LDA (GGA) calculations is 1.39 (1.33), suggesting that b-FeSi2 is brittle. The elastic constants and Debye temperature increase with increasing pressure.
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
References [1] Lange H. Phys Stat Sol B 1997;201:3. [2] Birkholz U, Grob E, Stöhrer U. In: Rowe DM, editor. Handbook of thermoelectrics. New York: CRC Press; 1995 [Chapter 24]. [3] Bost MC, Mahan JE. J Appl Phys 1985;58:2696. [4] Leong D, Harry M, Reeson KJ, Homewood KP. Nature 1997;387:686. [5] Maeda Y. Thin Solid Films 2007;515:8118. [6] Dusansoy Y, Protas J, Wandij R, Roques B. Acta Crystalogr B 1971;27:1209. [7] Takarabe K, Teranishi R, Oinuma J, Mori Y. J Phys Condens Matter 2002;14: 11007. [8] Takarabe K, Teranishi R, Oinuma J, Mori Y. Phys Rev B 2002;65:165215. [9] Takarabe K, Ikai T, Mori Y, Udono H, Kikuma I. J Appl Phys 2004;96:4903. [10] Mori Y, Ikai T, Teranishi R, Takarabe K. Phys Stat Sol B 2003;235:302. [11] Segall MD, Lindan PJD, Probert MJ, Pickard CJ, Hasnip PJ, Clark SJ, et al. J Phys Condens Matter 2002;14:2717. [12] Lin JS, Qteish A, Payne MC, Heine V. Phys Rev B 1993;47:4174. [13] Perdew JP, Zunger A. Phys Rev B 1981;23:5048. [14] Wang Y, Perdew JP. Phys Rev B 1991;44:13298. [15] Monkhorst HJ, Pack JD. Phys Rev B 1976;13:5188.
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
1227
CASTEP users guide. San Diego, CA: Accelrys Inc.; 2005 [and reference therein]. Sin’ko GV, Smirnov NA. J Phys Condens Matter 2002;14:6989. Mattesini M, Matar SF. Phys Rev B 2002;65:075110. Wu Z, Zhao E, Xiang H, Hao X, Liu X, Meng J. Phys Rev B 2007;76:054115. Voigt W. Lehrbuch der kristallphysik. Leipzig: Teubner; 1910 (reprinted 1928). Reuss A, Angew Z. Math Mech 1929;9:49. Hill R. Proc Phys Soc London Sect A 1952;65:349. Ravindran P, Fast L, Korzhavyi A, Johansson B, Wills J, Eriksson O. J Appl Phys 1998;84:4891. Christensen NE. Phys Rev B 1990;42:7148. van Ek J, Turchi PEA, Sterne PA. Phys Rev B 1996;54:7897. Moroni EG, Wolf W, Hafner J, Podloucky R. Phys Rev B 1999;59:12860. Pugh SF. Philos Mag 1954;45:823. Chung DH, Buessem WR. In: Vahldiek FW, Mersol SA, editors. Anisotropy in single crystal refractory compound, vol. 2. New York: Plenum; 1968. p. 217. Leibfried G, Ludwig W. Solid State Phys 1961;12:275. Anderson OL. J Phys Chem Solids 1963;24:909. Schreiber E, Anderson OL, Soga N. Elastic constants and their measurements. New York: McGraw-Hill; 1973. Waldecker G, Meinhold H, Birkholz U. Phys Stat Sol A 1973;15:143.
Contents lists available at ScienceDirect
Intermetallics journal homepage: www.elsevier.com/locate/intermet
First-principles calculations of the structural and elastic properties of b-FeSi2 at high-pressure Jun-ichi Tani*, Masanari Takahashi, Hiroyasu Kido Electronic Materials Research Division, Osaka Municipal Technical Research Institute, 1-6-50 Morinomiya, Joto-ku, Osaka 536-8553, Japan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 September 2009 Received in revised form 16 February 2010 Accepted 11 March 2010 Available online 3 April 2010
The pressure dependence of the structural and elastic properties of b-FeSi2 in the range 0e60 GPa was investigated using first-principles calculations based on density functional theory. Calculations were performed within the local density approximation as well as the generalized gradient approximation to the exchange correlation potential. The calculated lattice constants and internal parameters are in good agreement with previous experimental results. The nine independent elastic constants, c11, c22, c33, c44, c55, c66, c12, c13, and c23, of orthorhombic b-FeSi2 have been evaluated. The isotropic bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, elastic anisotropy, and Debye temperature of polycrystalline b-FeSi2 under pressure are also presented. Ó 2010 Elsevier Ltd. All rights reserved.
Keywords: A. Silicide, various B. Elastic properties B. Anisotropy E. Ab-initio calculations E. Phase stability, prediction
1. Introduction Transition metal silicides are of considerable interest from the viewpoint of their structural and functional applications [1]. The semiconducting phase of iron disilicide (b-FeSi2) has been studied as a material for thermoelectric conversion, solar cells, and optoelectronic applications [2e5]. Dusausoy et al. [6] reported the crystal structure of b-FeSi2 to be a base-centered orthorhombic system (space group: Cmca) having 48 atoms per unit cell with lattice constants a ¼ 0.9863 nm, b ¼ 0.7791 nm, and c ¼ 0.7833 nm Fig. 1 shows the unit cell of b-FeSi2. There are two crystallographically inequivalent sites for Fe and Si (FeI, FeII, SiI, and SiII). The unit cell contains 16 formula units distributed over 8 FeI, 8 FeII, 16 SiI, and 16 SiII. The two types of Fe sites are coordinated by 8 Si atoms with slightly different distances to Fe, and have the point symmetries 2 (C2) and m (Cs), respectively. Takarabe et al. [7e9] and Mori et al. [10] reported the pressure dependence of the structural and optical properties of b-FeSi2. They [9,10] pointed out that though a-FeSi2 is denser than b-FeSi2, its bulk modulus is only 75% that of b-FeSi2. Moreover, high-pressure X-ray diffraction measurements revealed that a high-pressure phase starts to appear at about 20 GPa. The elastic properties are very important to understanding solid-state physical, chemical, and mechanical
* Corresponding author. Tel.: þ81 6 6963 8081; fax: þ81 6 6963 8099. E-mail address: [email protected] (J.-i. Tani). 0966-9795/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2010.03.023
properties. Although the bulk modulus of b-FeSi2 has been investigated via theory and experiment, to the best of our knowledge, no other elastic constant of b-FeSi2 have been reported. Moreover, it is important to investigate the structural and elastic properties of b-FeSi2 at high-pressure so as to understand its phase stability. In the present study, to investigate b-FeSi2 on the first-principles, we performed calculations using the density functional pseudopotential method to obtain precise information on the structural and elastic properties of this system in the pressure range of 0e60 GPa. 2. Computational details Density functional theory (DFT) calculations were performed using the computer program CASTEP (Cambridge serial total energy package in material modeling, Accelrys Inc.) [11]. Here, we used the norm-conserving potential generated by means of the optimization scheme of Lin et al. [12]. The iron 3d64s2 orbitals and silicon 3s23p2 orbitals were treated as valence states. Two approximations for the exchange correlation potential were utilized, namely, the local density approximation (LDA) and the generalized gradient approximation (GGA). For the exchange correlation potentials, we used the CeperleyeAlder form of the LDA as parameterized by Perdew and Zunger [13], and for the GGA form we used the Perdew and Wang [14] interpolation formula, known as PW91. We constructed a base-centered orthorhombic primitive cell having 24 atoms (4 FeI, 4 FeII, 8 SiI, and 8 SiII). We expanded the
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
1223
Table 1 Structural properties of the calculated b-FeSi2, obtained from LDA as well as GGA calculations, in comparison with previous experimental data [6]. Deviations from the experimental data are shown in parentheses. Structural parameters
Calculated LDA
Cell parameters
a (Å) b (Å) c (Å) V (Å3) Atomic coordinates FeI (8d)
Fig. 1. Unit cell of b-FeSi2. The large spheres represent Fe and the small spheres represent Si. There are two crystallographically inequivalent sites of both Fe and Si (FeI, FeII, SiI, SiII).
valence electronic wave functions in a plane wave basis set up to an energy cutoff of 560 eV, which converges the total energy of the unit cell to better than 1 meV/atom. The density mixing method was used for electronic minimization. In the total energy calculations, integrations over the Brillouin zone were performed by using a 4 4 3 MonkhorstePack set [15], which produced 12 irreducible k points in the Brillouin zone of the 24-atom primitive cell. The structure was examined as a function of external hydrostatic pressure ranging from 0 to þ60 GPa. The elastic stiffness coefficients were determined using a finite strain technique [16] and a linear fit of the computed stress for small strains, where 6 strain amplitudes up to a maximum of 0.3% were used. 3. Results and discussion Table 1 compares the structural properties calculated for b-FeSi2 by the LDA and GGA calculations with previous experimental data [6]. The LDA results for lattice constants a, b, and c are 0.9%, 1.0%, and 0.9% smaller than the previous experimental results, which are within the range of typical LDA errors. The GGA results for lattice constants a, b, and c are in excellent agreement (within 0.2%) with the previous experimental results. In both LDA and GGA, the calculated internal parameters (x, y, z) of FeI, FeII, SiI, and SiII atoms are also in excellent agreement with the previous experimental results. Fig. 2 shows the pressure dependence of the normalized lattice parameters a/a0, b/b0, and c/c0 (a) as well as the normalized unit cell volume V/V0 (b) (where a0, b0, c0, and V0 are the equilibrium structural parameters at pressure P ¼ 0 GPa). The normalized
9.7734 7.7146 7.7663 585.56 0.2156 0 0 FeII (8f) 0.5 0.3085 0.1856 SiI (16g) 0.1285 0.2742 0.0516 SiII (16g) 0.3734 0.0456 0.2265
Experimental [6] GGA
(0.9%) (1.0%) (0.9%) (2.7%)
9.8692 7.7741 7.8455 601.94 0.2160 0 0 0.5 0.3082 0.1849 0.1282 0.2744 0.0516 0.3731 0.0457 0.2266
(þ0.1%) 9.863 (0.2%) 7.791 (þ0.2%) 7.833 (0.0%) 601.91 0.2146 0 0 0.5 0.3086 0.1851 0.1282 0.2746 0.0512 0.3727 0.0450 0.2261
volume V/V0 decreases with increasing pressure. The V/V0 values for b-FeSi2 at P ¼ 60 GPa from our LDA and GGA calculations are 0.814 and 0.802, respectively. The ratio of decline in LDA results is smaller than that in GGA results. The a/a0, b/b0, and c/c0 values for b-FeSi2 at P ¼ 60 GPa from our LDA (GGA) calculations are 0.923 (0.917), 0.943 (0.939), and 0.936 (0.931), respectively. We note that the order of compressibility is a-axis > c-axis > b-axis at P ¼ 0e60 GPa. Takarabe et al. [9] pointed out that the normalized linear compressibility of the a-axis is about 20%e30% greater than those of the c- and b-axes, indicating that the bonding along the a-axis is weaker than those along the other two axes. In order to understand the bonding, we calculated the nearest-neighbor distances of FeIeSiI, FeIeSiII, FeIIeSiI, FeIIeSiII, SiIeSiI, SiIeSiII, SiIIeSiII, and FeIeFeII of relaxed b-FeSi2 (Table 2). At P ¼ 60 GPa, the normalized nearest-neighbor distances of FeIeSiI, FeIeSiII, FeIIeSiI, FeIIeSiII, SiIeSiI, SiIeSiII, SiIIeSiII, and FeIeFeII are respectively 94.1% (93.6%), 94.0% (93.2%), 94.1% (93.6%), 94.3% (93.7%), 93.3% (92.8%), 92.9% (92.5%), 92.5% (92.0%), and 90.5% (90.1%) with respect to those at P ¼ 0 GPa. Therefore, the order of compressibility is FeeFe > SieSi > FeeSi. The directions of the FeeFe, SieSi, and FeeSi bonds in b-FeSi2 greatly affect the compressibility of the a-, b-, and c-axes. The internal parameters (x, y, z) for FeI, FeII, SiI, and SiII atoms of b-FeSi2 at P ¼ 0 GPa from our LDA calculations are respectively (0.2156, 0, 0), (0.5, 0.3085, 0.1856), (0.1285, 0.2742, 0.0516), and (0.3734, 0.0456, 0.2265), while those at P ¼ 60 GPa are respectively (0.2101, 0, 0), (0.5, 0.3141, 0.1796), (0.1308, 0.2767, 0.0594), and (0.3735, 0.0510, 0.2209). The internal parameters for FeI, FeII, SiI, and SiII are slightly affected by the pressure. Therefore, the bond bending in b-FeSi2 might slightly affect the compressibility of the a-, b-, and c-axes. Table 3 shows the pressure dependence of elastic stiffness (cij) of b-FeSi2 obtained from LDA and GGA calculations. The elastic stiffness tensor cij of orthorhombic crystals has nine independent components, c11, c22, c33, c44, c55, c66, c12, c13, and c23. For orthorhombic crystals, mechanical stability requires the elastic constants satisfy the well-known Born stability criteria [17e19]: cii P > 0 (i ¼ 1e6), (c11 þ c22 2c12 4P) > 0, (c11 þ c33 2c13 4P) > 0, (c22 þ c33 2c23 4P) > 0, (c11 þ c22 þ c33 þ 2c12 þ 2c13 þ 2c23 9P) > 0
(1)
The elastic constants in Table 3 satisfy all of these conditions, which suggests that all the calculated structures in the range of 0e60 GPa are elastically stable. Because no experimental or
1224
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227 Table 2 Mean nearest-neighbor distances of FeeSi, SieSi, and FeeFe of b-FeSi2 at P ¼ 0 and 60 GPa, obtained from LDA as well as GGA calculations. The value in parentheses represents the percentage of the distances at P ¼ 60 GPa with respect to those at P ¼ 0 GPa. Method
LDA
Pressure (GPa) Mean nearest-neighbor FeIeSiI FeIeSiII distance (Å) FeIIeSiI FeIIeSiII SiIeSiI SiIeSiII SiIIeSiII FeIeFeII
0 2.3318 2.3439 2.3651 2.3541 2.5230 2.5102 2.4571 2.9498
GV ¼
GGA 60 2.1948 2.2032 2.2257 2.2190 2.3536 2.3331 2.2738 2.6681
(94.1%) (94.0%) (94.1%) (94.3%) (93.3%) (92.9%) (92.5%) (90.5%)
0 2.3525 2.3656 2.3870 2.3764 2.5478 2.5323 2.4811 2.9784
60 2.2030 2.2042 2.2347 2.2277 2.3636 2.3427 2.2831 2.6831
(93.6%) (93.2%) (93.6%) (93.7%) (92.8%) (92.5%) (92.0%) (90.1%)
1 1 ðc þ c22 þ c33 c12 c13 c23 Þ þ ðc44 þ c55 þ c66 Þ 15 11 5 (3)
Reuss’s bulk modulus (BR) and Voigt’s bulk modulus (BV) of the orthorhombic system are given by equations (4) and (5):
BR ¼
1 ðs11 þ s22 þ s33 Þ þ 2ðs12 þ s13 þ s23 Þ
(4)
BV ¼
1 2 ðc þ c22 þ c33 Þ þ ðc12 þ c13 þ c23 Þ 9 11 9
(5)
In equations (2) and (4), the sij are the elastic compliance, which is converted from cij using the relation between cij and sij in the orthorhombic system. Hill [22] has proved that the Voigt and Reuss equations represent the upper and lower limits of the polycrystalline constants, and the arithmetic mean value of the Voigt and Reuss moduli, GH ¼ (GR þ GV)/2 and BH ¼ (BRþBV)/2, are taken to estimate the shear and bulk moduli. E and n can be calculated using Hill’s shear and the bulk moduli (GH and BH), and they are given by equations (6) and (7):
E ¼
9BH GH 3BH þ GH
(6)
n¼
3BH 2GH 2ð3BH þ GH Þ
(7)
The values of B, G, E, and n are shown in Table 4. The BH value for b-FeSi2 at P ¼ 0 GPa from our LDA (GGA) calculations is 194.5 Fig. 2. Normalized lattice constants a/a0, b/b0, and c/c0 (a) and normalized unit cell volume V/V0 (b) of b-FeSi2 as a function of pressure P. The results from LDA and GGA calculations are represented by the solid and dashed curves, respectively.
theoretical data for the cij of b-FeSi2 are available in previous reports, the present work is the first theoretical prediction of the value of cij. From our calculated cij results, the isotropic shear modulus G, bulk modulus B, Young’s modulus E, and Poisson’s ratio n of polycrystalline b-FeSi2 were determined using the VoigteReusseHill averaging scheme [20e22]. According to the Voigt and Reuss approximation, Reuss’s shear modulus (GR) and the Voigt’s shear modulus (GV) in the orthorhombic system are given by equations (2) and (3) [23]:
GR ¼
15 4ðs11 þ s22 þ s33 Þ 4ðs12 þ s13 þ s23 Þ þ 3ðs44 þ s55 þ s66 Þ (2)
(172.8) GPa, which is in good agreement with previously calculated values (199 GPa by Christensen [24], 203 GPa by van Ek et al. [25], and 182e209 GPa by Moroni et al. [26]), and is w20%e29% lower than a previous experimental value determined from high-pressure X-ray diffraction measurements (243.5 GPa) [9,10]. The calculated BH value for b-FeSi2 increases with increasing pressure, and at the pressure of 60 GPa, the BH value from our LDA (GGA) calculations is 416.8 (399.4) GPa. The BH values at 60 GPa are 2.1e2.3 times larger than those at P ¼ 0 GPa. Pugh [27] reported that the quotient of bulk to shear modulus (B/G) is an indication of the extent of the plastic range of a pure metal, so that a high B/G value is associated with ductility and a low B/G value with brittleness. The critical value that separates ductile and brittle materials is w1.75. The calculated BH/GH values at P ¼ 0 GPa for b-FeSi2 from our LDA (GGA) calculations is 1.39 (1.33), suggesting that b-FeSi2 is brittle. However, the calculated BH/GH values increase with increasing pressure and the value at 60 GPa from our LDA (GGA) calculations is 1.83 (1.86), suggesting that b-FeSi2 becomes more ductile with increasing pressure.
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
1225
Table 3 Pressure dependence of elastic stiffness (cij) of b-FeSi2 obtained from LDA and GGA calculations. P (GPa) 0
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
10 20 30 40 50 60
c11 (GPa)
c22 (GPa)
c33 (GPa)
c44 (GPa)
c55 (GPa)
c66 (GPa)
c12 (GPa)
c13 (GPa)
c23 (GPa)
360.8 314.6 418.1 376.3 478.8 446.0 532.3 490.4 578.2 533.6 637.1 590.3 672.6 640.5
393.9 355.8 473.6 424.7 535.3 491.6 597.8 551.8 655.6 611.6 710.9 644.2 767.3 721.8
397.2 362.5 477.1 419.6 526.0 482.3 584.8 545.3 643.0 603.6 697.2 658.5 746.7 705.7
144.8 126.4 165.2 148.3 182.5 168.5 201.6 183.1 219.9 201.9 233.8 218.3 250.3 232.8
131.8 127.0 154.2 141.9 170.7 157.2 181.6 169.7 194.0 181.8 202.1 190.8 214.5 205.5
142.8 142.0 164.8 152.7 172.5 164.5 184.2 178.3 194.4 192.2 207.5 196.6 214.2 206.6
102.0 95.8 130.8 126.1 156.8 156.4 184.8 182.3 211.1 206.7 238.7 234.3 264.9 261.3
85.1 77.7 114.8 102.4 137.3 132.4 162.5 155.7 185.8 182.5 213.6 206.7 233.8 230.5
114.2 89.4 143.0 126.2 173.6 158.5 204.2 188.3 234.1 219.8 261.4 247.8 290.2 276.2
The shear anisotropic factors are given by [23]
A1 ¼
4c44 c11 þ c33 2c13
for the f100g planes
(8)
A2 ¼
4c55 c22 þ c33 2c23
for the f010g planes
(9)
A3 ¼
4c66 c11 þ c22 2c12
for the f001g planes
(10)
The calculated values of A1, A2, and A3 for b-FeSi2 are given in Table 5. A value of 1 means completely isotropic properties, whereas values smaller or greater than 1 measure degrees of anisotropy. It can be seen that b-FeSi2 exhibits low anisotropy at P ¼ 0e60 GPa, because the values of A1, A2, and A3 are in the range of 0.91e1.12, which is close to 1. Using the Voigt and Reuss moduli, the percentage anisotropies in the compressibility and shear (AB and AG) are defined as [28]
AB ¼
BV BR 100 BV þ BR
(11)
AG ¼
GV GR 100 GV þ GR
(12)
A value of 0% implies completely isotropic properties, whereas 100% represents the maximum anisotropy. The calculated values of AB and AG for b-FeSi2 are given in Table 5. It can be seen that the
anisotropies in both compressibility and shear are also small, because the values of AB and AG are in the range of 0.09%e0.34%. The anisotropies of the linear bulk moduli along the a-, b-, and caxes are defined as [23]
dP L Ba ¼ a ¼ da 1þaþb
(13)
dP Ba ¼ a db
(14)
dP Ba Bc ¼ c ¼ b dc
(15)
Bb ¼ b
where
L ¼ c11 þ 2c12 a þ c22 a2 þ 2c13 b þ c33 b þ 2c23 ab 2
(16)
a ¼
ðc11 c12 Þðc33 c13 Þ ðc23 c13 Þðc11 c13 Þ ðc33 c13 Þðc22 c12 Þ ðc13 c23 Þðc12 c23 Þ
(17)
b¼
ðc22 c12 Þðc11 c13 Þ ðc11 c12 Þðc23 c12 Þ ðc22 c12 Þðc33 c13 Þ ðc12 c23 Þðc13 c23 Þ
(18)
The anisotropies of the bulk modulus along the a- and c-axes with respect to the b-axis can be written as [23]
ABa ¼
Ba ¼ a Bb
(19)
Table 4 Pressure dependence of isotropic shear modulus (G) and bulk modulus (B) of polycrystalline b-FeSi2 obtained from the elastic constants cij using the VoigteReusseHill averaging scheme. Young’s modulus (E) and Poisson’s ratio (n) are determined using the Hill’s shear and bulk moduli (GH and BH). P (GPa) 0 10 20 30 40 50 60
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
GR (GPa)
GV (GPa)
GH (GPa)
BR (GPa)
BV (GPa)
BH (GPa)
E (GPa)
n
BH/GH
140.2 129.8 161.8 145.9 176.3 162.6 190.5 176.5 203.9 190.6 216.6 200.6 227.8 214.9
140.6 130.4 162.2 146.3 176.6 162.9 191.0 177.0 204.7 191.2 217.4 201.4 229.0 215.6
140.4 130.1 162.0 146.1 176.5 162.7 190.8 176.8 204.3 190.9 217.0 201.0 228.4 215.3
194.1 172.5 237.0 213.4 273.6 256.3 311.4 291.9 346.5 327.5 383.9 361.9 415.4 398.3
194.9 173.2 238.4 214.4 275.1 257.2 313.1 293.3 348.8 329.6 385.8 363.4 418.3 400.4
194.5 172.8 237.7 213.9 274.3 256.7 312.2 292.6 347.6 328.6 384.9 362.7 416.8 399.4
339.5 312.0 396.0 357.0 435.9 403.0 475.5 441.4 512.6 479.7 548.0 509.0 579.4 547.4
0.209 0.199 0.222 0.222 0.235 0.238 0.246 0.249 0.254 0.257 0.263 0.266 0.268 0.272
1.39 1.33 1.47 1.47 1.56 1.58 1.64 1.66 1.71 1.73 1.78 1.81 1.83 1.86
1226
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
Table 5 Pressure dependence of shear anisotropic factors A1, A2, and A3; anisotropy in compressibility and shear AB and AG, respectively; linear bulk moduli Ba, Bb, and Bc along the a-, b-, and c-axes of the orthorhombic cell, respectively; and anisotropies of the bulk moduli ABa and ABc along the a- and c-axes with respect to the b-axis, respectively. P (GPa) 0
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
10 20 30 40 50 60
ABc ¼
A1
A2
A3
AB (%)
AG (%)
Ba (GPa)
Bb (GPa)
Bc(GPa)
ABa
ABc
0.985 0.969 0.993 1.004 1.000 1.016 1.018 1.011 1.035 1.046 1.031 1.045 1.052 1.052
0.937 0.942 0.928 0.959 0.956 0.957 0.938 0.942 0.934 0.938 0.913 0.946 0.919 0.939
1.037 1.186 1.046 1.113 0.985 1.053 0.969 1.053 0.958 1.051 0.953 1.027 0.941 0.984
0.22 0.21 0.30 0.25 0.26 0.18 0.28 0.25 0.32 0.33 0.25 0.21 0.34 0.27
0.13 0.24 0.12 0.14 0.09 0.09 0.14 0.12 0.19 0.15 0.20 0.19 0.25 0.18
515.1 457.6 612.1 561.7 715.5 688.3 808.8 760.7 886.3 833.0 997.6 945.0 1054.9 1027.4
642.5 569.6 793.5 730.3 929.7 866.3 1069.1 1002.2 1203.1 1137.0 1315.0 1202.8 1466.3 1388.2
604.5 538.4 754.6 650.7 846.5 772.1 961.7 897.7 1079.2 1027.2 1187.5 1144.7 1286.4 1223.7
0.802 0.803 0.771 0.769 0.770 0.795 0.756 0.759 0.737 0.733 0.759 0.786 0.719 0.740
0.941 0.945 0.951 0.891 0.911 0.891 0.900 0.896 0.897 0.903 0.903 0.952 0.877 0.882
a Bc ¼ b Bb
(20)
The calculated values of Ba, Bb, Bc, ABa, and ABc for b-FeSi2 are given in Table 5. At P ¼ 0 GPa, the directional bulk moduli Ba, Bb, and Bc from our LDA (GGA) calculations are 515.1 (457.6), 642.5 (569.6), and 604.5 (538.4) GPa, respectively. Therefore, the bulk modulus is the greatest along b-axis and the smallest along a-axis. As discussed earlier, the order of compressibility is a-axis > c-axis > b-axis. Therefore, the order of compressibility corresponds with the order of directional bulk moduli Ba < Bc < Bb. The directional bulk moduli, Ba, Bb, and Bc, increase with pressure. From the cij, we can calculate the elastic Debye temperature (QD). At T ¼ 0 K, QD equals the Debye temperature determined from specific heat measurements [29]. QD is proportional to the averaged sound velocity (vm) and can be calculated by the following equation [30]:
QD
h 3n NA r 1=3 ¼ vm k 4p M
(21)
where h is Planck’s constant, k Boltzmann’s constant, NA Avogadro’s number, r the density, M the molecular weight, and n the number of atoms per formula unit. Table 6 Pressure dependence of density (r), transverse (vt), longitudinal (vl), and average elastic velocities (vm), and Debye temperature (QD) calculated from elastic constants. P (GPa) 0 10 20 30 40 50 60
LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA LDA GGA
r (g/cm3)
vt (m/s)
vl (m/s)
vm (m/s)
QD (K)
5.082 4.944 5.335 5.211 5.547 5.443 5.742 5.648 5.921 5.835 6.087 6.006 6.242 6.165
5256 5130 5510 5295 5640 5468 5764 5594 5874 5720 5971 5785 6049 5909
8667 8369 9222 8856 9585 9329 9933 9672 10 233 9997 10 525 10 247 10 750 10 552
5808 5663 6098 5860 6251 6062 6396 6210 6525 6355 6639 6435 6730 6577
751.2 725.6 801.5 764.2 832.4 802.1 861.6 831.9 888.0 860.6 911.8 879.9 932.1 907.2
The average wave velocity (vm) in the polycrystalline material can be calculated using
vm
!#1=3 " 1 2 1 ¼ þ 3 v3t v3 l
(22)
where vt and vl are the traverse and longitudinal elastic wave velocities of the polycrystalline material, respectively. The values of the average traverse and longitudinal sound velocities (vt and vl) can be obtained using Hill’s shear modulus (GH) and the bulk modulus (BH) as follows [31]:
vt ¼
sffiffiffiffiffiffiffi GH
r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 r BH þ GH vl ¼ 3
(23)
(24)
The calculated values of r, vt, vl, vm, and QD for b-FeSi2 are given in Table 6. At P ¼ 0 GPa, QD of b-FeSi2 from our LDA (GGA) calculations is 752.1 (725.6) K. The value is w15%e20% higher than the previous experimental value of 630 K obtained from analysis of specific heat by Waldecker et al. [32]. The Debye temperature of bFeSi2 increases monotonically with increasing pressure. 4. Conclusions The pressure dependence of the structural and elastic properties of b-FeSi2 in the range 0e60 GPa was investigated using firstprinciples calculations based on density functional theory. The calculated elastic constants indicate that b-FeSi2 is elastically stable in the range of 0e60 GPa. The BH value for b-FeSi2 at P ¼ 0 GPa from our LDA (GGA) calculations is 194.5 (172.8) GPa, which is in good agreement with previously calculated values, and is w20%e29% lower than a previous experimental value determined from highpressure X-ray diffraction measurements (243.5 GPa). Further experimental studies will be necessary in order to settle this discrepancy between theoretical calculations and experimental results. The bulk modulus of b-FeSi2 is the greatest along b-axis and the smallest along a-axis. However, we predict that anisotropies in both compressibility and shear are small. The calculated BH/GH values at P ¼ 0 GPa for b-FeSi2 from our LDA (GGA) calculations is 1.39 (1.33), suggesting that b-FeSi2 is brittle. The elastic constants and Debye temperature increase with increasing pressure.
J.-i. Tani et al. / Intermetallics 18 (2010) 1222e1227
References [1] Lange H. Phys Stat Sol B 1997;201:3. [2] Birkholz U, Grob E, Stöhrer U. In: Rowe DM, editor. Handbook of thermoelectrics. New York: CRC Press; 1995 [Chapter 24]. [3] Bost MC, Mahan JE. J Appl Phys 1985;58:2696. [4] Leong D, Harry M, Reeson KJ, Homewood KP. Nature 1997;387:686. [5] Maeda Y. Thin Solid Films 2007;515:8118. [6] Dusansoy Y, Protas J, Wandij R, Roques B. Acta Crystalogr B 1971;27:1209. [7] Takarabe K, Teranishi R, Oinuma J, Mori Y. J Phys Condens Matter 2002;14: 11007. [8] Takarabe K, Teranishi R, Oinuma J, Mori Y. Phys Rev B 2002;65:165215. [9] Takarabe K, Ikai T, Mori Y, Udono H, Kikuma I. J Appl Phys 2004;96:4903. [10] Mori Y, Ikai T, Teranishi R, Takarabe K. Phys Stat Sol B 2003;235:302. [11] Segall MD, Lindan PJD, Probert MJ, Pickard CJ, Hasnip PJ, Clark SJ, et al. J Phys Condens Matter 2002;14:2717. [12] Lin JS, Qteish A, Payne MC, Heine V. Phys Rev B 1993;47:4174. [13] Perdew JP, Zunger A. Phys Rev B 1981;23:5048. [14] Wang Y, Perdew JP. Phys Rev B 1991;44:13298. [15] Monkhorst HJ, Pack JD. Phys Rev B 1976;13:5188.
[16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
1227
CASTEP users guide. San Diego, CA: Accelrys Inc.; 2005 [and reference therein]. Sin’ko GV, Smirnov NA. J Phys Condens Matter 2002;14:6989. Mattesini M, Matar SF. Phys Rev B 2002;65:075110. Wu Z, Zhao E, Xiang H, Hao X, Liu X, Meng J. Phys Rev B 2007;76:054115. Voigt W. Lehrbuch der kristallphysik. Leipzig: Teubner; 1910 (reprinted 1928). Reuss A, Angew Z. Math Mech 1929;9:49. Hill R. Proc Phys Soc London Sect A 1952;65:349. Ravindran P, Fast L, Korzhavyi A, Johansson B, Wills J, Eriksson O. J Appl Phys 1998;84:4891. Christensen NE. Phys Rev B 1990;42:7148. van Ek J, Turchi PEA, Sterne PA. Phys Rev B 1996;54:7897. Moroni EG, Wolf W, Hafner J, Podloucky R. Phys Rev B 1999;59:12860. Pugh SF. Philos Mag 1954;45:823. Chung DH, Buessem WR. In: Vahldiek FW, Mersol SA, editors. Anisotropy in single crystal refractory compound, vol. 2. New York: Plenum; 1968. p. 217. Leibfried G, Ludwig W. Solid State Phys 1961;12:275. Anderson OL. J Phys Chem Solids 1963;24:909. Schreiber E, Anderson OL, Soga N. Elastic constants and their measurements. New York: McGraw-Hill; 1973. Waldecker G, Meinhold H, Birkholz U. Phys Stat Sol A 1973;15:143.