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Lattice dynamics of β-FeSi2 from first-principles calculations
ARTICLE IN PRESS Physica B 405 (2010) 2200–2204
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Lattice dynamics of b-FeSi2 from first-principles calculations 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 in f o
a b s t r a c t
Article history: Received 14 January 2010 Accepted 2 February 2010
The structure, lattice dynamics, and some thermodynamic properties of b-FeSi2 were investigated using first-principles calculations that are based on density functional theory. The fully relaxed structure parameters of b-FeSi2 are in good agreement with previous experimental data. The linear response method is applied in order to determine the phonon dispersion relations, phonon density of states, and the Born effective charge. The computed thermodynamic quantities such as vibrational entropy, specific heat, and Debye temperature are in agreement with previous experimental data. & 2010 Elsevier B.V. All rights reserved.
Keywords: b-FeSi2 Phonons Thermodynamic properties Born effective charge
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 [2–5]. 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 = 0.9863 nm, b=0.7791 nm, and c= 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. As several physical properties of b-FeSi2 at finite temperatures depend on its lattice-dynamical behavior, it is important to understand the vibrational properties of b-FeSi2. However, experimental and theoretical studies on the phonons of b-FeSi2 have been very limited. Guizzetti et al. [7] reported that the anisotropy effect can be observed much more clearly in the vibrational spectra of b-FeSi2 than in its electronic spectra, owing to the enhanced sensitivity of the IR and Raman features to local structural distortions. Walterfang et al. [8] investigated the atomic vibrational density of states (VDOS) of b-FeSi2 experimentally and theoretically. They reported that the measured partial VDOS measured by nuclear resonant inelastic X-ray scattering (NRIXS) showed good agreement with the theoretical value of VODS of b-FeSi2; this theoretical value was computed using the Corresponding author. Tel.: + 81 6 6963 8081; fax: + 81 6 6963 8099.
E-mail address: [email protected] (J.-i. Tani). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.02.008
density functional theory combined with the direct method. To the best of our knowledge, detailed first-principles lattice dynamical calculation data on b-FeSi2 such as phonon dispersion relations, Born effective charges, and thermodynamic properties such as vibrational entropy, specific heat, and Debye temperature have not been reported. In the present study, we used the density functional theory combined with the linear response method [9–11] for performing first-principles calculations in order to determine the structure, lattice dynamics, Born effective charges, and some thermodynamic quantities of b-FeSi2. These computed thermodynamic quantities were then compared with those obtained in previous experiments.
2. Computational details Density functional theory (DFT) calculations within the pseudopotential and local density approximations (LDA) were performed using the computer program CASTEP (Cambridge serial total energy package in materials studio, Accelrys Inc.) [12]. Here, we used the norm-conserving potential generated by means of the optimization scheme of Lin et al. [13]. The iron 3d64s2 orbitals and silicon 3s23p2 orbitals were treated as valence states. For the exchange correlation potential, we used the Ceperley–Alder form of the LDA as parameterized by Perdew and Zunger [14]. We constructed a base-centered orthorhombic primitive cell having 24 atoms (4 FeI, 4 FeII, 8 SiI, and 8 SiII). We expanded the 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
ARTICLE IN PRESS J.-i. Tani et al. / Physica B 405 (2010) 2200–2204
Fe-I
Table 1 Comparison of the structural properties of b-FeSi2 calculated in this study with those obtained in previous experiments [6].
Fe-II
Structural parameters Cell parameters
Si-II
Atomic coordinates
Y Z
2201
Calculated ˚ a (A) ˚ b (A)
9.7734 ( 0.9%)
9.863
7.7146 ( 1.0%)
7.791
˚ c (A) V (A˚ 3)
7.7663 ( 0.9%)
FeI (8d)
Si-I FeII (8f)
X
SiI (16g)
Fe-II Si-I
SiII (16g)
Si-II
Experimental [6]
585.56 ( 2.7%) x y z x y z x y z x y z
0.2156 0 0 0.5 0.3085 0.1856 0.1285 0.2742 0.0516 0.3734 0.0456 0.2265
7.833 601.91 0.2146 0 0 0.5 0.3086 0.1851 0.1282 0.2746 0.0512 0.3727 0.0450 0.2261
Deviations from the experimental data are shown in parentheses.
Fe-I
Y Z
X
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, and SiII).
by using a 4 4 3 Monkhorst-Pack set [15], which produced 12 irreducible k points in the Brillouin zone of the 24-atom primitive cell. The phonons were calculated from the optimized geometry by employing the linear response method. They were calculated by using an energy cutoff of 560 eV and a 3 3 3 MonkhorstPack set, which produced 8 irreducible k points in the irreducible Brillouin zone and included the determination of the longitudinal optical/transverse optical (LO/TO) splitting. The phonon frequencies are converged to better than 1 cm 1.
3. Results and discussion
Fig. 2. Calculated phonon dispersion curves in b-FeSi2 along several lines of high symmetry in Brillouin zone. Frequencies at the G point are shown in Table 2.
Table 1 shows a comparison of the structural properties of
b-FeSi2 calculated in this study with those obtained in previous experiments [6]. The calculated lattice constants a, b, and c are 0.9%, 1.0%, and 0.9% smaller than those obtained in the previous experiments; this variation is within the range of typical LDA errors. 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 calculated phonon dispersion relations of bFeSi2. G, Z, T, Y, S, and R are the high-symmetry points. The phonon frequencies are obtained at all these high-symmetry points and at other intermediate points. Because no experimental or theoretical data for the phonon dispersion of b-FeSi2 are available in previous reports, the present study is the first theoretical prediction of the phonon dispersion. When the G point is approached from different directions (q-0 along G Z, G Y, and G S), the IR active phonon mode branches (B1u, B2u, and B3u) approach different limiting values at G (q = 0) owing to the crystal anisotropy of b-FeSi2. Table 2 shows a comparison between the phonon frequencies of b-FeSi2 at the G point calculated in this study and those
obtained from previous experimental data. Since the primitive cell contains 24 atoms, a factor group analysis (where the crystal point group is D2h in the Schoenflies notation) leads to 72 vibrational modes at the G point, of which 3 are acoustic modes and 69 are optical modes [16,17]. The optical phonons at the G point of the Brillouin zone can be classified as
Gac : B1u þB2u þB3u ;
ð1Þ
Gopt : 9Ag ðRÞ þ 9B1g ðRÞ þ 9B2g ðRÞ þ 9B3g ðRÞ þ9B1u ðIÞ þ 9B2u ðIÞ þ 7B3u ðIÞ þ 8Au ;
ð2Þ
where R and I are the Raman-active and IR-active modes, respectively. The mode of Au is Raman- and IR-inactive (silent mode). Except for the Raman frequencies (197, 253, and 346 cm 1 [17], 194.0 and 247.3 cm 1 [23]) in the A1g mode, the symmetry assignment of IR and Raman frequencies has not been proposed yet. Therefore, a comparison between the calculated and experimental frequencies is difficult as a large number of phonon modes would have to be considered. The calculated A1g
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Table 2 Comparison between the phonon frequencies (in cm 1) and symmetry assignment of b-FeSi2 at the G point calculated in this study and those obtained from previous experimental data [17–23]. Calc. (This work) Infrared B1u(1) B1u(2) B1u(3) B1u(5) B1u(6) B1u(7) B1u(8) B1u(9)
203/203 288/293 304/305 362/379 381/396 401/418 439/439 445/500
B2u(1) B2u(2) B2u(3) B2u(5) B2u(6) B2u(7) B2u(8) B2u(9)
270/270 301/302 307/308 353/354 359/364 445/451 452/491 497/499
B3u(1) B3u(2) B3u(3) B3u(5) B3u(6) B3u(7)
230/230 255/255 274/284 354/363 363/445 478/479
Raman B1g(1) B1g(2) B1g (3) B1g (4) B1g (5) B1g (6) B1g (7) B1g (8) B1g (9)
185 198 240 285 317 324 366 385 428
B2g(1) B2g (2) B2g (3) B2g (4) B2g (5) B2g (6) B2g (7) B2g (8) B2g (9)
179 254 295 326 345 350 383 402 420
B3g(1) B3g(2) B3g(3) B3g(4) B3g(5) B3g(6) B3g(7) B3g(8) B3g(9)
212 236 297 341 386 401 431 458 466
Ag(1) Ag(2) Ag(3) Ag(4) Ag(5) Ag(6) Ag(7) Ag(8) Ag(9)
208 210 257 264 352 398 415 464 517
Silent Au(1) Au(5)
191 355
Au(2) Au(6)
251 380
Au(3) Au(7)
254 425
Au(4) Au(8)
304 463
Exp. [17]
Exp. [18]
Exp. [19]
Exp. [20]
Exp. [21]
259a 263b 267c 310a 316b 315c 324c 345a 360b 360c 375c
263.8 300 310 425
263 294 308 425
216 263 294 345 392 425 472
265/275 308/329 347/383 446/485
176a 179c 195a(Ag) 197c(Ag) 200a 206c 247a(Ag) 253c(Ag) 346c(Ag)
173 192.3 198 245
Exp. [22]
Exp. [23]
171 190 199 247
190.6 194.0 (Ag) 199.6 227.1 231.6 247.3 (Ag) 254.3 274.1 281.2
311.8 325.8 339.5 370.7 386.2 388.2 400.4 442.6 446.3
Two numbers in a row correspond to TO/LO frequencies. a
˚ on Si (111) substrate b-FeSi2 film (2000A)
b
Polycrystalline bulk samples
c
b-FeSi2 film (1 mm) on FeSi (110) substrate
phonon frequencies of 208, 254, and 352 cm 1 are observed to be in fairly good agreement with previous experimental data within 7% error. The IR-active phonon mode branches (B1u, B2u, and B3u) split into traverse optical (TO) and longitudinal optical (LO) phonons with different frequencies due to the macroscopic electric fields associated with the LO phonons. The LO–TO splitting is computed on the basis of the Born effective charge [24]. The calculated 3 3 Born effective charge tensors (Z*) for FeI, FeII, SiI, and SiII are as follows: 0 1 6:93 0:00 0:00 B C ð3Þ Z ðFeI Þ ¼ @ 0:00 6:77 0:64 A; 0:00 0:35 6:89 0
5:75
B Z ðFeII Þ ¼ @ 0:00 0:00 0
þ 3:15 B Z ðSiI Þ ¼ @ 0:35 0:18
0
þ 3:19 B Z ðSiII Þ ¼ @ þ 0:10 0:21
0:00
1
0:00
7:17
þ0:12 C A;
þ0:25
6:97
þ 0:13
þ0:29
þ 3:65
þ0:21 C A;
1
þ 0:18
þ 3:47
þ 0:21
þ0:29
þ 3:32
þ0:15 C A:
0:35
ð4Þ
ð5Þ
1 ð6Þ
þ 3:46
The Z* for FeI, FeII, SiI, and SiII has off-diagonal elements due to the low symmetry in the orthorhombic phase. The diagonal elements of Z* for FeI, FeII, SiI, and SiII are not equal, i.e., a Zyy a Zzz . The average values of the diagonal elements of Z* Zxx are 6.86e, 6.63e, + 3.42e, and + 3.32e for FeI, FeII, SiI, and SiII, respectively. These values are much larger than the values of the Mulliken charges of b-FeSi2 ( 0.11e, 0.12e, +0.07e, and + 0.05e
Fig. 3. Total phonon DOS in b-FeSi2. Partial phonon DOS of constituent elements (FeI, FeII, SiI, and SiII) per atom are also shown.
for FeI, FeII, SiI, and SiII, respectively). It is known that anomalous Z* values are due to changes that occur in the covalent bonding as a result of atomic motions. Therefore, the anomalous Z* may be
ARTICLE IN PRESS J.-i. Tani et al. / Physica B 405 (2010) 2200–2204
2203
interpreted in terms of mixed covalent-ionic bonding due to the hybridization of Si 3p and Fe 3d orbitals. Fig. 3 shows the total phonon DOS of b-FeSi2 and of its partial components per atom. In a frequency region below 12 THz, the phonon DOS is dominated by the movements of Si atoms as well as those of Fe atoms. However, in a frequency region above 12 THz, the phonon DOS is dominated by Si atoms. The partial phonon DOS of b-FeSi2 clearly indicates that Si atoms have higher frequencies than Fe atoms since Fe atoms are heavier than Si atoms. In spite of two crystallographically inequivalent sites for Fe and Si (FeI, FeII, SiI, and SiII), the partial phonon DOS between FeI and FeII atoms as well as that between SiI and SiII atoms does not differ much. The values of the partial phonon DOS of FeI and FeII atoms calculated in our study are in good agreement with the experimental values of Fe-projected VDOS obtained using NRIXS by Walterfang et al. [8]. The calculated value of the total phonon DOS was used to evaluate the temperature dependence of thermodynamic quantities of b-FeSi2 in a quasi-harmonic approximation. The temperature dependence of the vibrational entropy, S, is given by 8 ‘o
ð9Þ
where a is the volume thermal expansion, V is the molar volume, and B is the bulk modulus. For b-FeSi2, the calculated V= 22.04 cm3/mol and B= 194.5 GPa at 0 K [25]. a is calculated using the experimental thermal expansion coefficient b ((a: 16.697+ 0.001442T, b: 0.026 +0.001673T, and c: 0.415 + 0.001872T) 10 6 K 1) [1]. Cv in the Debye model is given by [26] Z T 3 YD=T x4 ex CvD ðTÞ ¼ 9NkB dx; ð10Þ YD ðex 1Þ2 0 where N is the number of atoms per cell. Thus, the value of the Debye temperature, YD, at a given temperature, T, is obtained by first calculating the actual heat capacity using Eq. (8) and then inverting Eq. (10). Figs. 4(a), (b), and (c) show the calculated thermodynamic quantities (S, Cv, Cp and YD) of b-FeSi2 in the temperature range 0–1000 K. In Figs. 4(a) and (b), the calculated values of S and Cp are compared with those obtained from previous experimental data. S is calculated to be 55.7 J/mol K at 298 K, while the value of S obtained in earlier experiments is 51.4 J/mol K [27]. Therefore, the calculated value of S is in fairly good agreement with the experimental data. The Cp value of b-FeSi2 has been measured by Krentsis et al. [28] and by Maglic and Parrot [29]. For To50 K, no experimental data are available. The calculated value of Cp is in good agreement with the experimental value obtained by Maglic
Fig. 4. Calculated thermodynamic quantities of b-FeSi2 in comparison with previous experimental data. (a) vibrational entropy (S), (b) specific heat (Cv, Cp), and (c) Debye temperature (YD). Solid lines are calculated results. A circle symbol in (a) and dashed lines in (b) are the experimental results quoted from Refs. [27–29]. Dot line in (b) is the limiting Dulong–Petit value of 9R (74.8J/mol K).
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and Parrot, and is slightly lower than the experimental value obtained by Krentsis et al. At high temperatures, the calculated Cv will approach the limiting Dulong–Petit value of 9R (74.8 J/mol K). The YD value of b-FeSi2 is calculated to be 620–660 K in the temperature range 100–1000 K. This value is in good agreement with the previous experimental value (630 K) obtained from analysis of specific heat by Waldecker et al. [30]. The Debye frequency, nD, is given by
nD ¼
kB YD h
ð11Þ
On the basis of the calculated value of YD, i.e., 620–660 K, the Debye frequency is estimated to be 12.9–13.8 THz.
4. Conclusions The structure, lattice dynamics, and some thermodynamic properties of b-FeSi2 were investigated using first-principles calculations that are based on density functional theory. The fully relaxed structure parameters of b-FeSi2 are in good agreement with previous experimental data. The linear response method is applied in order to determine the phonon dispersion relations, phonon density of states, and the Born effective charge. The computed thermodynamic quantities such as vibrational entropy, specific heat, and Debye temperature are in agreement with previous experimental data. References [1] H. Lange, Phys. Status Solidi B 201 (1997) 3. ¨ in: D.M. Rowe (Ed.), Handbook of Thermo[2] U. Birkholz, E. Grob, U. Stohrer, electrics, CRC Press, New York, 1995 Chapter 24.
[3] [4] [5] [6] [7] [8]
[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
M.C. Bost, J.E. Mahan, J. Appl. Phys. 58 (1985) 2696. D. Leong, M. Harry, K.J. Reeson, K.P. Homewood, Nature 387 (1997) 686. Y. Maeda, Thin Solid Films 515 (2007) 8118. Y. Dusansoy, J. Protas, R. Wandij, B. Roques, Acta Crystalogr. B 27 (1971) 1209. G. Guizzetti, F. Marabelli, M. Patrini, P. Pellegrino, B. Pivac, L. Miglio, V. Meregalli, H. Lange, W. Henrion, V. Tomm, Phys. Rev. B 21 (1997) 14290. M. Walterfang, W. Keune, E. Schuster, A.T. Zayak, P. Entel, W. Sturhahn, T.S. Toellner, E.E. Alp, P.T. Jochym, K. Parlinski, Phys. Rev. B 71 (2005) 035309. X. Gonze, Phys. Rev. B 55 (1997) 10337. X. Gonze, C. Lee, Phys. Rev. B 55 (1997) 10355. K. Refson, S.J. Clark, P.R. Tulip, Phys. Rev. B 73 (2006) 155114. S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M.C. Payne, Z. Kristallogr. 220 (2005) 567. J.S. Lin, A. Qteish, M.C. Payne, V. Heine, Phys. Rev. B 47 (1993) 4174. J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. D.L. Rousseau, R.P. Bauman, S.P.S. Porto, J. Raman Spectrosc. 10 (1981) 253. K. Lefki, P. Muret, E. Bustarret, N. Boutarek, R. Madar, J. Cheverier, J. Derriem, M. Brunel, Solid State Commun 80 (1991) 791. R. Ayache, A. Bouabellou, E. Richter, Mater. Sci. Semicond. Process. 7 (2004) 463. V. Daraktchieva, M. Baleva, E. Goranava, Ch. Angelov, Vacuum 58 (2000) 415. W.Z. Shen, S.C. Shen, W.G. Tang, L.W. Wang, J. Appl. Phys. 78 (1995) 4793. U. Birkholz, H. Finkenrath, J. Naegele, N. Uhle, Phys. Status Solidi 30 (1968) K81. H. Kakemoto, Y. Makita, S. Sakuragi, T. Tsukamoto, Jpn. J. Appl. Phys. Part 1 38 (1999) 5192. Y. Maeda, H. Udono, Y. Terai, Thin Solid Films 461 (2004) 165. M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford, 1954. J. Tani, M. Takahashi, H. Kido, unpublished work. N.W. Achcroft, N.D. Mermin, Solid State Phys., Saunders College, Philadelphia, 1976. National Bureau of Standards, Technical Notes No. 270-4 (US Government Printing Office, Washington, 1969). P.G. Krentsis, P.V. Geld, G.I. Kalishevitch, Techernaya Metall. 11 (1963) 146. ¨ Bildung und Wissenschaft, K. Maglic, J.E. Parrot, Bundesministerium fur Forshungsbericht (1970) K70-01. G. Waldecker, H. Meinhold, U. Birkholz, Phys. Status Solidi A 15 (1973) 143.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Lattice dynamics of b-FeSi2 from first-principles calculations 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 in f o
a b s t r a c t
Article history: Received 14 January 2010 Accepted 2 February 2010
The structure, lattice dynamics, and some thermodynamic properties of b-FeSi2 were investigated using first-principles calculations that are based on density functional theory. The fully relaxed structure parameters of b-FeSi2 are in good agreement with previous experimental data. The linear response method is applied in order to determine the phonon dispersion relations, phonon density of states, and the Born effective charge. The computed thermodynamic quantities such as vibrational entropy, specific heat, and Debye temperature are in agreement with previous experimental data. & 2010 Elsevier B.V. All rights reserved.
Keywords: b-FeSi2 Phonons Thermodynamic properties Born effective charge
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 [2–5]. 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 = 0.9863 nm, b=0.7791 nm, and c= 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. As several physical properties of b-FeSi2 at finite temperatures depend on its lattice-dynamical behavior, it is important to understand the vibrational properties of b-FeSi2. However, experimental and theoretical studies on the phonons of b-FeSi2 have been very limited. Guizzetti et al. [7] reported that the anisotropy effect can be observed much more clearly in the vibrational spectra of b-FeSi2 than in its electronic spectra, owing to the enhanced sensitivity of the IR and Raman features to local structural distortions. Walterfang et al. [8] investigated the atomic vibrational density of states (VDOS) of b-FeSi2 experimentally and theoretically. They reported that the measured partial VDOS measured by nuclear resonant inelastic X-ray scattering (NRIXS) showed good agreement with the theoretical value of VODS of b-FeSi2; this theoretical value was computed using the Corresponding author. Tel.: + 81 6 6963 8081; fax: + 81 6 6963 8099.
E-mail address: [email protected] (J.-i. Tani). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.02.008
density functional theory combined with the direct method. To the best of our knowledge, detailed first-principles lattice dynamical calculation data on b-FeSi2 such as phonon dispersion relations, Born effective charges, and thermodynamic properties such as vibrational entropy, specific heat, and Debye temperature have not been reported. In the present study, we used the density functional theory combined with the linear response method [9–11] for performing first-principles calculations in order to determine the structure, lattice dynamics, Born effective charges, and some thermodynamic quantities of b-FeSi2. These computed thermodynamic quantities were then compared with those obtained in previous experiments.
2. Computational details Density functional theory (DFT) calculations within the pseudopotential and local density approximations (LDA) were performed using the computer program CASTEP (Cambridge serial total energy package in materials studio, Accelrys Inc.) [12]. Here, we used the norm-conserving potential generated by means of the optimization scheme of Lin et al. [13]. The iron 3d64s2 orbitals and silicon 3s23p2 orbitals were treated as valence states. For the exchange correlation potential, we used the Ceperley–Alder form of the LDA as parameterized by Perdew and Zunger [14]. We constructed a base-centered orthorhombic primitive cell having 24 atoms (4 FeI, 4 FeII, 8 SiI, and 8 SiII). We expanded the 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
ARTICLE IN PRESS J.-i. Tani et al. / Physica B 405 (2010) 2200–2204
Fe-I
Table 1 Comparison of the structural properties of b-FeSi2 calculated in this study with those obtained in previous experiments [6].
Fe-II
Structural parameters Cell parameters
Si-II
Atomic coordinates
Y Z
2201
Calculated ˚ a (A) ˚ b (A)
9.7734 ( 0.9%)
9.863
7.7146 ( 1.0%)
7.791
˚ c (A) V (A˚ 3)
7.7663 ( 0.9%)
FeI (8d)
Si-I FeII (8f)
X
SiI (16g)
Fe-II Si-I
SiII (16g)
Si-II
Experimental [6]
585.56 ( 2.7%) x y z x y z x y z x y z
0.2156 0 0 0.5 0.3085 0.1856 0.1285 0.2742 0.0516 0.3734 0.0456 0.2265
7.833 601.91 0.2146 0 0 0.5 0.3086 0.1851 0.1282 0.2746 0.0512 0.3727 0.0450 0.2261
Deviations from the experimental data are shown in parentheses.
Fe-I
Y Z
X
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, and SiII).
by using a 4 4 3 Monkhorst-Pack set [15], which produced 12 irreducible k points in the Brillouin zone of the 24-atom primitive cell. The phonons were calculated from the optimized geometry by employing the linear response method. They were calculated by using an energy cutoff of 560 eV and a 3 3 3 MonkhorstPack set, which produced 8 irreducible k points in the irreducible Brillouin zone and included the determination of the longitudinal optical/transverse optical (LO/TO) splitting. The phonon frequencies are converged to better than 1 cm 1.
3. Results and discussion
Fig. 2. Calculated phonon dispersion curves in b-FeSi2 along several lines of high symmetry in Brillouin zone. Frequencies at the G point are shown in Table 2.
Table 1 shows a comparison of the structural properties of
b-FeSi2 calculated in this study with those obtained in previous experiments [6]. The calculated lattice constants a, b, and c are 0.9%, 1.0%, and 0.9% smaller than those obtained in the previous experiments; this variation is within the range of typical LDA errors. 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 calculated phonon dispersion relations of bFeSi2. G, Z, T, Y, S, and R are the high-symmetry points. The phonon frequencies are obtained at all these high-symmetry points and at other intermediate points. Because no experimental or theoretical data for the phonon dispersion of b-FeSi2 are available in previous reports, the present study is the first theoretical prediction of the phonon dispersion. When the G point is approached from different directions (q-0 along G Z, G Y, and G S), the IR active phonon mode branches (B1u, B2u, and B3u) approach different limiting values at G (q = 0) owing to the crystal anisotropy of b-FeSi2. Table 2 shows a comparison between the phonon frequencies of b-FeSi2 at the G point calculated in this study and those
obtained from previous experimental data. Since the primitive cell contains 24 atoms, a factor group analysis (where the crystal point group is D2h in the Schoenflies notation) leads to 72 vibrational modes at the G point, of which 3 are acoustic modes and 69 are optical modes [16,17]. The optical phonons at the G point of the Brillouin zone can be classified as
Gac : B1u þB2u þB3u ;
ð1Þ
Gopt : 9Ag ðRÞ þ 9B1g ðRÞ þ 9B2g ðRÞ þ 9B3g ðRÞ þ9B1u ðIÞ þ 9B2u ðIÞ þ 7B3u ðIÞ þ 8Au ;
ð2Þ
where R and I are the Raman-active and IR-active modes, respectively. The mode of Au is Raman- and IR-inactive (silent mode). Except for the Raman frequencies (197, 253, and 346 cm 1 [17], 194.0 and 247.3 cm 1 [23]) in the A1g mode, the symmetry assignment of IR and Raman frequencies has not been proposed yet. Therefore, a comparison between the calculated and experimental frequencies is difficult as a large number of phonon modes would have to be considered. The calculated A1g
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Table 2 Comparison between the phonon frequencies (in cm 1) and symmetry assignment of b-FeSi2 at the G point calculated in this study and those obtained from previous experimental data [17–23]. Calc. (This work) Infrared B1u(1) B1u(2) B1u(3) B1u(5) B1u(6) B1u(7) B1u(8) B1u(9)
203/203 288/293 304/305 362/379 381/396 401/418 439/439 445/500
B2u(1) B2u(2) B2u(3) B2u(5) B2u(6) B2u(7) B2u(8) B2u(9)
270/270 301/302 307/308 353/354 359/364 445/451 452/491 497/499
B3u(1) B3u(2) B3u(3) B3u(5) B3u(6) B3u(7)
230/230 255/255 274/284 354/363 363/445 478/479
Raman B1g(1) B1g(2) B1g (3) B1g (4) B1g (5) B1g (6) B1g (7) B1g (8) B1g (9)
185 198 240 285 317 324 366 385 428
B2g(1) B2g (2) B2g (3) B2g (4) B2g (5) B2g (6) B2g (7) B2g (8) B2g (9)
179 254 295 326 345 350 383 402 420
B3g(1) B3g(2) B3g(3) B3g(4) B3g(5) B3g(6) B3g(7) B3g(8) B3g(9)
212 236 297 341 386 401 431 458 466
Ag(1) Ag(2) Ag(3) Ag(4) Ag(5) Ag(6) Ag(7) Ag(8) Ag(9)
208 210 257 264 352 398 415 464 517
Silent Au(1) Au(5)
191 355
Au(2) Au(6)
251 380
Au(3) Au(7)
254 425
Au(4) Au(8)
304 463
Exp. [17]
Exp. [18]
Exp. [19]
Exp. [20]
Exp. [21]
259a 263b 267c 310a 316b 315c 324c 345a 360b 360c 375c
263.8 300 310 425
263 294 308 425
216 263 294 345 392 425 472
265/275 308/329 347/383 446/485
176a 179c 195a(Ag) 197c(Ag) 200a 206c 247a(Ag) 253c(Ag) 346c(Ag)
173 192.3 198 245
Exp. [22]
Exp. [23]
171 190 199 247
190.6 194.0 (Ag) 199.6 227.1 231.6 247.3 (Ag) 254.3 274.1 281.2
311.8 325.8 339.5 370.7 386.2 388.2 400.4 442.6 446.3
Two numbers in a row correspond to TO/LO frequencies. a
˚ on Si (111) substrate b-FeSi2 film (2000A)
b
Polycrystalline bulk samples
c
b-FeSi2 film (1 mm) on FeSi (110) substrate
phonon frequencies of 208, 254, and 352 cm 1 are observed to be in fairly good agreement with previous experimental data within 7% error. The IR-active phonon mode branches (B1u, B2u, and B3u) split into traverse optical (TO) and longitudinal optical (LO) phonons with different frequencies due to the macroscopic electric fields associated with the LO phonons. The LO–TO splitting is computed on the basis of the Born effective charge [24]. The calculated 3 3 Born effective charge tensors (Z*) for FeI, FeII, SiI, and SiII are as follows: 0 1 6:93 0:00 0:00 B C ð3Þ Z ðFeI Þ ¼ @ 0:00 6:77 0:64 A; 0:00 0:35 6:89 0
5:75
B Z ðFeII Þ ¼ @ 0:00 0:00 0
þ 3:15 B Z ðSiI Þ ¼ @ 0:35 0:18
0
þ 3:19 B Z ðSiII Þ ¼ @ þ 0:10 0:21
0:00
1
0:00
7:17
þ0:12 C A;
þ0:25
6:97
þ 0:13
þ0:29
þ 3:65
þ0:21 C A;
1
þ 0:18
þ 3:47
þ 0:21
þ0:29
þ 3:32
þ0:15 C A:
0:35
ð4Þ
ð5Þ
1 ð6Þ
þ 3:46
The Z* for FeI, FeII, SiI, and SiII has off-diagonal elements due to the low symmetry in the orthorhombic phase. The diagonal elements of Z* for FeI, FeII, SiI, and SiII are not equal, i.e., a Zyy a Zzz . The average values of the diagonal elements of Z* Zxx are 6.86e, 6.63e, + 3.42e, and + 3.32e for FeI, FeII, SiI, and SiII, respectively. These values are much larger than the values of the Mulliken charges of b-FeSi2 ( 0.11e, 0.12e, +0.07e, and + 0.05e
Fig. 3. Total phonon DOS in b-FeSi2. Partial phonon DOS of constituent elements (FeI, FeII, SiI, and SiII) per atom are also shown.
for FeI, FeII, SiI, and SiII, respectively). It is known that anomalous Z* values are due to changes that occur in the covalent bonding as a result of atomic motions. Therefore, the anomalous Z* may be
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2203
interpreted in terms of mixed covalent-ionic bonding due to the hybridization of Si 3p and Fe 3d orbitals. Fig. 3 shows the total phonon DOS of b-FeSi2 and of its partial components per atom. In a frequency region below 12 THz, the phonon DOS is dominated by the movements of Si atoms as well as those of Fe atoms. However, in a frequency region above 12 THz, the phonon DOS is dominated by Si atoms. The partial phonon DOS of b-FeSi2 clearly indicates that Si atoms have higher frequencies than Fe atoms since Fe atoms are heavier than Si atoms. In spite of two crystallographically inequivalent sites for Fe and Si (FeI, FeII, SiI, and SiII), the partial phonon DOS between FeI and FeII atoms as well as that between SiI and SiII atoms does not differ much. The values of the partial phonon DOS of FeI and FeII atoms calculated in our study are in good agreement with the experimental values of Fe-projected VDOS obtained using NRIXS by Walterfang et al. [8]. The calculated value of the total phonon DOS was used to evaluate the temperature dependence of thermodynamic quantities of b-FeSi2 in a quasi-harmonic approximation. The temperature dependence of the vibrational entropy, S, is given by 8 ‘o
ð9Þ
where a is the volume thermal expansion, V is the molar volume, and B is the bulk modulus. For b-FeSi2, the calculated V= 22.04 cm3/mol and B= 194.5 GPa at 0 K [25]. a is calculated using the experimental thermal expansion coefficient b ((a: 16.697+ 0.001442T, b: 0.026 +0.001673T, and c: 0.415 + 0.001872T) 10 6 K 1) [1]. Cv in the Debye model is given by [26] Z T 3 YD=T x4 ex CvD ðTÞ ¼ 9NkB dx; ð10Þ YD ðex 1Þ2 0 where N is the number of atoms per cell. Thus, the value of the Debye temperature, YD, at a given temperature, T, is obtained by first calculating the actual heat capacity using Eq. (8) and then inverting Eq. (10). Figs. 4(a), (b), and (c) show the calculated thermodynamic quantities (S, Cv, Cp and YD) of b-FeSi2 in the temperature range 0–1000 K. In Figs. 4(a) and (b), the calculated values of S and Cp are compared with those obtained from previous experimental data. S is calculated to be 55.7 J/mol K at 298 K, while the value of S obtained in earlier experiments is 51.4 J/mol K [27]. Therefore, the calculated value of S is in fairly good agreement with the experimental data. The Cp value of b-FeSi2 has been measured by Krentsis et al. [28] and by Maglic and Parrot [29]. For To50 K, no experimental data are available. The calculated value of Cp is in good agreement with the experimental value obtained by Maglic
Fig. 4. Calculated thermodynamic quantities of b-FeSi2 in comparison with previous experimental data. (a) vibrational entropy (S), (b) specific heat (Cv, Cp), and (c) Debye temperature (YD). Solid lines are calculated results. A circle symbol in (a) and dashed lines in (b) are the experimental results quoted from Refs. [27–29]. Dot line in (b) is the limiting Dulong–Petit value of 9R (74.8J/mol K).
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and Parrot, and is slightly lower than the experimental value obtained by Krentsis et al. At high temperatures, the calculated Cv will approach the limiting Dulong–Petit value of 9R (74.8 J/mol K). The YD value of b-FeSi2 is calculated to be 620–660 K in the temperature range 100–1000 K. This value is in good agreement with the previous experimental value (630 K) obtained from analysis of specific heat by Waldecker et al. [30]. The Debye frequency, nD, is given by
nD ¼
kB YD h
ð11Þ
On the basis of the calculated value of YD, i.e., 620–660 K, the Debye frequency is estimated to be 12.9–13.8 THz.
4. Conclusions The structure, lattice dynamics, and some thermodynamic properties of b-FeSi2 were investigated using first-principles calculations that are based on density functional theory. The fully relaxed structure parameters of b-FeSi2 are in good agreement with previous experimental data. The linear response method is applied in order to determine the phonon dispersion relations, phonon density of states, and the Born effective charge. The computed thermodynamic quantities such as vibrational entropy, specific heat, and Debye temperature are in agreement with previous experimental data. References [1] H. Lange, Phys. Status Solidi B 201 (1997) 3. ¨ in: D.M. Rowe (Ed.), Handbook of Thermo[2] U. Birkholz, E. Grob, U. Stohrer, electrics, CRC Press, New York, 1995 Chapter 24.
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