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First-principles calculations for structure and equation of state of MgB2 at high pressure
ARTICLE IN PRESS
Physica B 370 (2005) 281–286 www.elsevier.com/locate/physb
First-principles calculations for structure and equation of state of MgB2 at high pressure Xiang-Rong Chena,b,, Hai-Yan Wanga, Yan Chenga, Yan-Jun Haoa a Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, PR China International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China
b
Received 19 June 2005; received in revised form 8 September 2005; accepted 15 September 2005
Abstract We investigate the structure and the equation of state of compound MgB2 at high pressure using the full-potential linearized muffin-tin orbital scheme within the generalized gradient approximation correction in the frame of density functional theory. Through the quasi-harmonic Debye model, in which the phononic effects are considered, we have obtained successfully the bulk modulus and the thermal expansion of MgB2. r 2005 Elsevier B.V. All rights reserved. PACS: 64.30.+t; 71.15.Mb; 71.15.Mb; 74.70.Ad Keywords: Equation of state; Full-potential linearized muffin-tin orbital; Generalized gradient approximation; MgB2
MgB2, an intermetallic compound, has been known since 1953. In 2001, the discovery of the superconductivity in compound MgB2 with a large superconducting temperature T c of 39 K [1] has attracted considerable experimentalists as well as theoreticians, and has taken the whole materials science community by surprise. Since then, a great number of interesting results have been reported for the understanding of the superconductivity in Corresponding author. Institute of Atomic and Molecular
Physics, Sichuan University, Chengdu 610065, PR China. Tel.: +86 288 540 5516; fax: +86 288 540 5515. E-mail addresses: [email protected], [email protected] (X.-R. Chen).
MgB2. [2–16]. For example, Cava [2] has described MgB2 as Genie in a bottle. The behaviors of MgB2 in high magnetic field have been studied by Larbalestier et al. [3]. The magnesium isotope effect reported by Bud’ko et al. [4] and Hinks et al. [5] strongly suggests it to be a phonon-mediated superconductor, and the electron-phonon coupling will play an important role in the superconductivity of the intermetallic compound. The pressure dependence of the superconducting transition temperature, T c , has been investigated by Saito et al. [13] and by Monteverde et al. [14], who reported a decrease of T c under pressure with a higher decrease rate than for conventional superconductors.
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.09.025
ARTICLE IN PRESS 282
X.-R. Chen et al. / Physica B 370 (2005) 281–286
The nature of bonding in boron layer in MgB2 depends crucially on the lattice constants a and c, and the changes of a and c will greatly affect the electronic structures around the Fermi energy E F . Therefore, it is very important to optimize a and c theoretically when investigating the properties of MgB2 under high pressure, such as the superconductivity and the equation of state (EOS). The EOS is the key thermodynamic properties of a solid, and determines the solid behavior with respect to changes in the macroscopic variables, such as pressure (P) and temperature (T). The structural properties of materials under high-pressure will provide us fundamental changes in bonding nature. Since the pressure effect can influence the electronic band in different directions of Brillouin zone (BZ) and hence physical properties, to investigate the EOS from first-principles calculations is the main objective for both physics and chemistry of crystals. In this work, we focus on the properties of compound MgB2 under high pressure. We apply the full-potential linearized muffin-tin orbital (FP-LMTO) scheme [17] within the generalized gradient approximation correction (GGA) in the frame of density functional theory [18] to calculate the lattice parameters and the EOS of MgB2 via the lmtART program [19]. In the FP-LMTO method, we have adopted the generalized gradient approximation for the exchange-correlation functions proposed by Perdewet al. [20]. The base geometry in this computational method consists of muffin-tin (MT) spheres and intertial parts. The basis set is comprised of augmented linear muffin-tin orbitals [21]. The MT sphere radii rMT are chosen as 2.922 and 1.687 a.u. for Mg atom and B atom, respectively. Inside the MT spheres, the basis functions, charge density and potential are expanded in symmetry adapted spherical harmonic functions together with a radial function. Fourier series are used in the interstitial regions. In the present calculations, the K-space integration has been performed using 6 6 6 k-points in the irreducible Brillouin zone. The charge density and basis functions were calculated out exactly in the MT spheres up to the angular momentum components l max ¼ 6. The final convergence is within 107 Ryd.
To investigate the thermodynamic properties of compound MgB2, we apply the quasi-harmonic Debye model, in which the phononic effects are considered. In the quasi-harmonic Debye model, the non-equilibrium Gibbs function G ðV ; P; TÞ is in the form of [22] G ðV ; P; TÞ ¼ EðV Þ þ PV þ AVib ðV ; TÞ.
(1)
On the right-hand side of Eq. (1), E (V) is the total energy per unit cell for MgB2, PV corresponds to the constant hydrostatic pressure condition, Avib is the vibrational Helmholtz free energy. Considering the quasi-harmonic approximation [23] and using the Debye model of phonon density of states, one can write Avib as [24–26] AVib ðY; TÞ 9Y Y=T þ 3 lnð1 e ¼ nKT Þ DðY=TÞ , ð2Þ 8T where YðV Þ is the Debye temperature, DðY=TÞ represents the Debye integral, and n is the number of atoms per formula unit. For an isotropic solid with Poisson ratio s ¼ 0:25 [27], Y is expressed by rffiffiffiffiffiffi _ BS 2 1=2 1=3 Y ¼ ½6p V n f ðsÞ , (3) K M where M is the molecular mass per formula unit, Bs is the adiabatic bulk modulus [22]. f ðsÞ and BS are given by Ref. [25]. Thus, the non-equilibrium Gibbs function G ðV ; P; TÞ, as a function of ðV ; P; TÞ, can be minimized with respect to volume V n qG ðV ; P; TÞ ¼ 0. (4) qV P;T And then, one can get the thermal EOS of compound MgB2 by solving Eq. (4). Since the experimental ratio c=a (a and c are the lattice parameters for a hexagonal crystal structure) of MgB2 is about 1.141 [1,6], we have calculated a series of different c=a ratios between 1.126 and 1.161, including 1.126, 1.130, 1.134, 1.138, 1.142, 1.146, 1.150, 1.154, 1.158, and 1.161. For a fixed c=a ratio, a series of different values of lattice constant a are set to calculate the total energies E and the corresponding volumes V , and then an energy–volume (E2V ) curve can be
ARTICLE IN PRESS X.-R. Chen et al. / Physica B 370 (2005) 281–286
obtained by fitting the calculated E2V data to the fourth-order natural strain EOS [28], in which the pressure–volume relationship expanded as V0 P ¼ 3B0 f N ½1 þ 32ðB0 2Þf N V þ 32ð1 þ B0 B00 þ ðB0 2Þ þ ðB0 2Þ2 Þf 2N ,
ð5Þ
where f N ¼ lnðl=l 0 Þ, which may be written as f N ¼ 1=3 lnðV =V 0 Þ for hydrostatic compression. B and B0 are hydrostatic bulk modulus and zero pressure bulk modulus, respectively. For truncation at third-order in the strain, the implied value of B00 is given by B00 ¼
1 ½1 þ ðB0 2Þ þ ðB0 2Þ2 . B0
(6)
283
The c=a –volume–energy relationship is shown in Fig. 1. We find that the hcp MgB2 with the ratio c=a of about 1.138 has the minimum energy, and the corresponding equilibrium lattice parameters a and c are about 3.093 and 3.520 A˚, respectively. In Table 1, we list our calculated results, together with the experimental data and other theoretical results. The calculated zero pressure bulk modulus B0 and the pressure derivation of bulk modulus B0 from the natural strain EOS are about 154.31 GPa and 3.63, respectively. Applying the quasi-harmonic Debye model to MgB2, we have obtained B0 ¼ 153:61 GPa and B0 ¼ 3:41, which are also listed in Table 1. These results are consistent with the experimental data [1,6] and those from other theoretical calculations [6–10]. However, since MgB2 is diverse and anisotropic [9,11], it is believed that the calculated results will be more accurate when the anisotropy of MgB2 is considered. One can define the bulk modulus Ba along the a-axis and Bc along the caxis as follows: Ba ¼ a
dp L ¼ , da 2 þ a
(7)
Bc ¼ c
dp Ba ¼ , dc a
(8)
L ¼ 2ðC 11 þ C 12 Þ þ 4C 13 a þ C 33 a2 ,
Fig. 1. The c=a—volume–energy relationship.
a¼
C 11 þ C 12 2C 13 . C 33 C 13
(9) (10)
Table 1 The calculated structural parameters compared with experiments and other theoretical results a (A˚)
c (A˚)
c/a
B0 (GPa)
B0
B00 (GPa1)
Present work
3.093
3.520
1.138
3.63a 3.41b
0.034 0.029
Vogt et al. [6] Singh [7] Islam et al. [8] Osorio-Guillen et al. [9] Ravindran et al. [10] Experiments [1,6]
3.089 3.048 3.064 3.085 3.080 3.086
3.548 3.487 3.493 3.557 3.532 3.521
1.149 1.144 1.140 1.153 1.147 1.141
154.31a 153.61b 150c 139710
a
Obtained from the natural strain EOS [28]. Obtained from the quasi-harmonic Debye model [22]. c Obtained from Eq. (11). b
122 151
3.4 3.65
15175
4.0
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284
Applying the calculated elastic constants at zero pressure, i.e. C 11 ¼ 462, C 12 ¼ 67, C 13 ¼ 41, C 33 ¼ 254, and C 44 ¼ 80 GPa [29], we obtain the bulk modulus Ba and Bc are 615.04, 293.07 GPa, respectively, that is, the compression along the aaxis is more difficult than that along the c-axis. This phenomenon can be understood by the knowledge of the bonding situations in MgB2, which is characterized as a strong covalent bonding in the boron layers, ionized magnesium atoms and a weaker interlayer metallic bonding. These mechanical properties suggest the anisotropy of MgB2. From the following formula of the zero-pressure bulk modulus B0 : B0 ¼
C 33 ðC 11 þ C 12 Þ 2C 213 , C 11 þ C 12 4C 13 þ 2C 33
(11)
we can obtain B0 ¼ 150 GPa. In the case, there is no constraint on the c=a dependence on lattice strain [30]. The calculated result agrees excellently with the experimental data [6]. In Fig. 2, we illustrate the pressure P dependence of the normalized volume V n ð¼ V =V 0 Þ of MgB2 in the range of 20–50 GPa, together with the experimental data at P ¼ 0240 GPa [31] and the theoretical results by Islam et al. [8]. The results obtained in this work consist with the experimental data [31], and seem to be better than
Fig. 3. Pressure dependence of the normalized lattice parameters and c=a ratio of MgB2.
those by Islam et al. [8]. The normalized lattice parameters a=a0 , c=c0 and the ratio c=a as a function of pressure are plotted in Fig. 3, where a0 and c0 are their values at T ¼ 0 and P ¼ 0, respectively. By fitting the calculated data to second-order polynomials, we obtain the following relationships at T ¼ 0 K : a=a0 ¼ 1:000 0:00190P þ 1:46159 105 P2 , (12) c=c0 ¼ 1:000 0:00342P þ 2:65743 105 P2 , (13) c=a ¼ 1:138 0:00172P þ 1:20368 105 P2 . (14)
Fig. 2. The normalized volume V n as a function of pressure P. The solid lines with square, circular and trigonal symbols represent the present work, experimental data and the other theoretical results obtained by Islam et al. [8], respectively.
These results agree with those obtained by Jorgensen et al. [32] through the neutron diffraction measurements under low pressure (o 0.62 GPa), also agree with those measured using a Merrill–Bassett diamond anvil cell up to 8 GPa [6]. It is shown that, as pressure increase, the equilibrium ratio c=a ratio decrease. The compression along the c-axis is much larger than that along the a-axis, consistent with the comparatively weaker (Mg–B) bonds that determine the c-axis length. The obtained isothermal V n 2P and isobaric V 2T relationship curves in the wide range of 0–240 GPa are illustrated in Figs. 4 and 5,
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285
respectively. For the isothermal compressions shown in Fig. 4, the volume is greatly affected by both the pressure and the temperature. The effect of increasing pressure on MgB2 is the same as decreasing temperature of MgB2. On the other hand, as the pressure increases, the volume compression decreases at a given temperature, and the volume compression at high temperature is less than that at lower temperature under a given pressure. As for the isobaric curves shown in Figs. 5(a) and (b), we find that, under lower pressure, the volume varies quickly as the temperature rise. Under higher pressure, it becomes moderate, and the V 2T relations are nearly linear. The V 2T relation at P ¼ 0 GPa (Fig. 5(a)) in the range of temperatures 0–300 K is consistent with the neutron powder diffraction measurements [32]. The compression behaviors of MgB2 correspond to the bonding situations in MgB2. When pressure increases, the atoms in the interlayers become closer, and the interactions between these atoms become stronger. Compared to K3C60, a fairly soft material with d ln V =dP ¼ 0:036 GPa1 [33], MgB2 is a tightly packed incompressible solid with d ln V =dP ¼ 0:0076 GPa1 . MgB2 is less compressible in the basal plane, in which the covalent B–B bonds lie. The interlayer linear compressibility (d ln c=dP ¼ 0:00353 GPa1 ) is about 1.8 times larger than that the in-plane value (d ln a=dP ¼ 0:00196 GPa1 ).
Fig. 5. The volume–temperature relationship curves: (a) under lower pressures, and (b) under higher pressures.
Fig. 4. The volume compressions V n of MgB2 up to pressure 240 GPa at the temperatures of 300, 1000, and 2000 K, respectively.
In summary, we apply the full-potential linearized muffin-tin orbital (FP-LMTO) scheme within the generalized gradient approximation to calculate the structure and equation of state of MgB2 at high pressure. It is demonstrated that the ratio c=a of about 1.138 is the most stable structure for hcp MgB2, the calculated zero pressure bulk module B0 and the pressure derivation of bulk modulus B00 from the natural strain EOS and from the quasiharmonic Debye model are about 154.31 GPa and 3.63, 153.61 GPa and 3.41, respectively. Through the quasi-harmonic Debye model, the dependences of the volume compression on pressure and the volume on temperature are successfully obtained. The authors would like to thank the support by the National Natural Science Foundation of
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China under Grant no. 10576020, and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry under Grant no.2004176-6-4. References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J.A. Kimitsu, Nature 410 (2001) 63. [2] R.J. Cava, Nature 410 (2001) 23. [3] D.C. Larbalestier, L.D. Cooley, M.O. Rikel, A.A. Polyanskil, J. Jiang, S. Patnaik, X.Y. Cai, D.M. Feldmann, A. Gurevich, A.A. Squitieri, M.T. Naus, C.B. Eom, E.E. Hellstrom, R.J. Cava, K.A. Regan, N. Rogado, M.A. Hayward, T. He, J.S. Slusky, P. Khalifah, K. Inumaru, M. Haas, Nature 410 (2001) 186. [4] S.L. Bud’ko, G. Lapertot, C. Petrovic, C.E. Cunningham, N. Anderson, P.C. Canfield, Phys. Rev. Lett. 86 (2001) 1877. [5] D.G. Hinks, H. Claus, J.D. Jorgensen, Nature (London) 411 (2001) 457. [6] T. Vogt, G. Schneider, J.A. Hriljac, G. Yang, J.S. Abell, Phys. Rev. B 63 (2001) 220505(R). [7] P.P. Singh, Phys. Rev. Lett. 87 (2001) 087004. [8] A.K.M.A. Islam, F.N. Islam, S.H. Rose, J. Phys.: Condens. Matter 13 (2001) L641. [9] J.M. Osorio-Guillen, S.I. Simak, Y. Wang, B. Johansson, R. Ahuja, Solid State Commun. 123 (2002) 257. [10] P. Ravindran, P. Vajeeston, R. Vidya, A. Kjekshus, H. Fjellvag, Phys. Rev. B 64 (2001) 224509. [11] F. Parvin, A.K.M.A. Islam, F.N. Islam, et al., Physica C 390 (2003) 16. [12] H.Z. Guo, X.R. Chen, J. Zhu, L.C. Cai, J. Gao, Chin. Phys. Lett. 22 (2005) 1764. [13] E. Satio, T. Taknenobu, T. Ito, Y. Iwasa, K. Prassides, T. Arima, J. Phys.: Condens. Matter 13 (2001) L267. [14] M. Monteverde, M. Nunez-Regueiro, K.A. Regan, M.A. Hayward, T. He, S.M. Loureiro, R.J. Cava, Science 92 (2001) 75. [15] L. Chen, H. Li, Chin. Phys. Lett. 20 (2003) 1128.
[16] X.R. Chen, H.Z. Guo, L.C. Cai, J. Gao, Chin. Phys. Lett. 22 (2005) 1504. [17] J.M. Wills, O. Eriksson, M. Alouani, D.L. Price, in: H. Dreysse (Ed.), Electronic Structure and Physical Properties of Solids, Springer, Berlin, 2000, p. 148; J.M. Wills, B.R. Cooper, Phys. Rev. B 36 (1987) 3809; D.L. Price, B.R. Cooper, Phys. Rev. B 39 (1989) 4945. [18] P. Hohenberg, W. Kohn, Phys. Rev.B 136 (1964) 864; W. Kohn, L.J. Sham, Phys. Rev.A 140 (1965) 1133. [19] S.Y. Savrasov, Phys. Rev. B 54 (1996) 16 470. [20] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [21] O.K. Andersen, Phys. Rev. B 12 (1975) 3060. [22] M.A. Blanco, E. Francisco, V. Luana, Comput. Phys. Commun. 158 (2004) 57. [23] A.A. Maradudin, E.W. Montroll, G.H. Weiss, I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Academic Press, New York, 1971. [24] M.A. Blanco, A. Martı´ n Penda´s, E. Francisco, et al., J. Mol. Struct. (Theochem.) 368 (1996) 245. [25] E. Francisco, J.M. Recio, M.A. Blanco, A. Martı´ n Penda´s, J. Phys. Chem. 102 (1998) 1595; E. Francisco, M.A. Blanco, G. Sanjurjo, Phys. Rev. B 63 (2001) 094 107. [26] M. Flo´rez, J.M. Recio, E. Francisco, M.A. Blanco, A.M. Penda´s, Phy. Rev. B 66 (2002) 144 112. [27] J.P. Poirier, Introduction to the Physics of the Earth’s Interior, Cambridge University, England, 1991. [28] J.P. Poirier, A. Tarantola, Phys. Earth Planet Int. 109 (1998) 1. [29] H.Y. Wang, J.Z. Zhao, Z.L. Liu, X.F. Li, J. Atomic Mol. Phys. 22 (2005) 226. [30] G. Steinle-Neumann, L. Stixrude, R.E. Cohen, Phys. Rev. B 60 (1999) 791. [31] P. Bordet, M. Mezouar, Phys. Rev. B 64 (2001) 172502. [32] I.D. Jorgensen, D.G. Hinks, S. Short, Phys. Rev. B 63 (2001) 224522. [33] O. Zhou, G.B.M. Vaughan, Q. Zhu, J.E. Ficher, P.A. Heiney, N. Coustel, J.P. McCauley, A.B. Smith, Science 255 (1992) 835.
Physica B 370 (2005) 281–286 www.elsevier.com/locate/physb
First-principles calculations for structure and equation of state of MgB2 at high pressure Xiang-Rong Chena,b,, Hai-Yan Wanga, Yan Chenga, Yan-Jun Haoa a Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, PR China International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, PR China
b
Received 19 June 2005; received in revised form 8 September 2005; accepted 15 September 2005
Abstract We investigate the structure and the equation of state of compound MgB2 at high pressure using the full-potential linearized muffin-tin orbital scheme within the generalized gradient approximation correction in the frame of density functional theory. Through the quasi-harmonic Debye model, in which the phononic effects are considered, we have obtained successfully the bulk modulus and the thermal expansion of MgB2. r 2005 Elsevier B.V. All rights reserved. PACS: 64.30.+t; 71.15.Mb; 71.15.Mb; 74.70.Ad Keywords: Equation of state; Full-potential linearized muffin-tin orbital; Generalized gradient approximation; MgB2
MgB2, an intermetallic compound, has been known since 1953. In 2001, the discovery of the superconductivity in compound MgB2 with a large superconducting temperature T c of 39 K [1] has attracted considerable experimentalists as well as theoreticians, and has taken the whole materials science community by surprise. Since then, a great number of interesting results have been reported for the understanding of the superconductivity in Corresponding author. Institute of Atomic and Molecular
Physics, Sichuan University, Chengdu 610065, PR China. Tel.: +86 288 540 5516; fax: +86 288 540 5515. E-mail addresses: [email protected], [email protected] (X.-R. Chen).
MgB2. [2–16]. For example, Cava [2] has described MgB2 as Genie in a bottle. The behaviors of MgB2 in high magnetic field have been studied by Larbalestier et al. [3]. The magnesium isotope effect reported by Bud’ko et al. [4] and Hinks et al. [5] strongly suggests it to be a phonon-mediated superconductor, and the electron-phonon coupling will play an important role in the superconductivity of the intermetallic compound. The pressure dependence of the superconducting transition temperature, T c , has been investigated by Saito et al. [13] and by Monteverde et al. [14], who reported a decrease of T c under pressure with a higher decrease rate than for conventional superconductors.
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.09.025
ARTICLE IN PRESS 282
X.-R. Chen et al. / Physica B 370 (2005) 281–286
The nature of bonding in boron layer in MgB2 depends crucially on the lattice constants a and c, and the changes of a and c will greatly affect the electronic structures around the Fermi energy E F . Therefore, it is very important to optimize a and c theoretically when investigating the properties of MgB2 under high pressure, such as the superconductivity and the equation of state (EOS). The EOS is the key thermodynamic properties of a solid, and determines the solid behavior with respect to changes in the macroscopic variables, such as pressure (P) and temperature (T). The structural properties of materials under high-pressure will provide us fundamental changes in bonding nature. Since the pressure effect can influence the electronic band in different directions of Brillouin zone (BZ) and hence physical properties, to investigate the EOS from first-principles calculations is the main objective for both physics and chemistry of crystals. In this work, we focus on the properties of compound MgB2 under high pressure. We apply the full-potential linearized muffin-tin orbital (FP-LMTO) scheme [17] within the generalized gradient approximation correction (GGA) in the frame of density functional theory [18] to calculate the lattice parameters and the EOS of MgB2 via the lmtART program [19]. In the FP-LMTO method, we have adopted the generalized gradient approximation for the exchange-correlation functions proposed by Perdewet al. [20]. The base geometry in this computational method consists of muffin-tin (MT) spheres and intertial parts. The basis set is comprised of augmented linear muffin-tin orbitals [21]. The MT sphere radii rMT are chosen as 2.922 and 1.687 a.u. for Mg atom and B atom, respectively. Inside the MT spheres, the basis functions, charge density and potential are expanded in symmetry adapted spherical harmonic functions together with a radial function. Fourier series are used in the interstitial regions. In the present calculations, the K-space integration has been performed using 6 6 6 k-points in the irreducible Brillouin zone. The charge density and basis functions were calculated out exactly in the MT spheres up to the angular momentum components l max ¼ 6. The final convergence is within 107 Ryd.
To investigate the thermodynamic properties of compound MgB2, we apply the quasi-harmonic Debye model, in which the phononic effects are considered. In the quasi-harmonic Debye model, the non-equilibrium Gibbs function G ðV ; P; TÞ is in the form of [22] G ðV ; P; TÞ ¼ EðV Þ þ PV þ AVib ðV ; TÞ.
(1)
On the right-hand side of Eq. (1), E (V) is the total energy per unit cell for MgB2, PV corresponds to the constant hydrostatic pressure condition, Avib is the vibrational Helmholtz free energy. Considering the quasi-harmonic approximation [23] and using the Debye model of phonon density of states, one can write Avib as [24–26] AVib ðY; TÞ 9Y Y=T þ 3 lnð1 e ¼ nKT Þ DðY=TÞ , ð2Þ 8T where YðV Þ is the Debye temperature, DðY=TÞ represents the Debye integral, and n is the number of atoms per formula unit. For an isotropic solid with Poisson ratio s ¼ 0:25 [27], Y is expressed by rffiffiffiffiffiffi _ BS 2 1=2 1=3 Y ¼ ½6p V n f ðsÞ , (3) K M where M is the molecular mass per formula unit, Bs is the adiabatic bulk modulus [22]. f ðsÞ and BS are given by Ref. [25]. Thus, the non-equilibrium Gibbs function G ðV ; P; TÞ, as a function of ðV ; P; TÞ, can be minimized with respect to volume V n qG ðV ; P; TÞ ¼ 0. (4) qV P;T And then, one can get the thermal EOS of compound MgB2 by solving Eq. (4). Since the experimental ratio c=a (a and c are the lattice parameters for a hexagonal crystal structure) of MgB2 is about 1.141 [1,6], we have calculated a series of different c=a ratios between 1.126 and 1.161, including 1.126, 1.130, 1.134, 1.138, 1.142, 1.146, 1.150, 1.154, 1.158, and 1.161. For a fixed c=a ratio, a series of different values of lattice constant a are set to calculate the total energies E and the corresponding volumes V , and then an energy–volume (E2V ) curve can be
ARTICLE IN PRESS X.-R. Chen et al. / Physica B 370 (2005) 281–286
obtained by fitting the calculated E2V data to the fourth-order natural strain EOS [28], in which the pressure–volume relationship expanded as V0 P ¼ 3B0 f N ½1 þ 32ðB0 2Þf N V þ 32ð1 þ B0 B00 þ ðB0 2Þ þ ðB0 2Þ2 Þf 2N ,
ð5Þ
where f N ¼ lnðl=l 0 Þ, which may be written as f N ¼ 1=3 lnðV =V 0 Þ for hydrostatic compression. B and B0 are hydrostatic bulk modulus and zero pressure bulk modulus, respectively. For truncation at third-order in the strain, the implied value of B00 is given by B00 ¼
1 ½1 þ ðB0 2Þ þ ðB0 2Þ2 . B0
(6)
283
The c=a –volume–energy relationship is shown in Fig. 1. We find that the hcp MgB2 with the ratio c=a of about 1.138 has the minimum energy, and the corresponding equilibrium lattice parameters a and c are about 3.093 and 3.520 A˚, respectively. In Table 1, we list our calculated results, together with the experimental data and other theoretical results. The calculated zero pressure bulk modulus B0 and the pressure derivation of bulk modulus B0 from the natural strain EOS are about 154.31 GPa and 3.63, respectively. Applying the quasi-harmonic Debye model to MgB2, we have obtained B0 ¼ 153:61 GPa and B0 ¼ 3:41, which are also listed in Table 1. These results are consistent with the experimental data [1,6] and those from other theoretical calculations [6–10]. However, since MgB2 is diverse and anisotropic [9,11], it is believed that the calculated results will be more accurate when the anisotropy of MgB2 is considered. One can define the bulk modulus Ba along the a-axis and Bc along the caxis as follows: Ba ¼ a
dp L ¼ , da 2 þ a
(7)
Bc ¼ c
dp Ba ¼ , dc a
(8)
L ¼ 2ðC 11 þ C 12 Þ þ 4C 13 a þ C 33 a2 ,
Fig. 1. The c=a—volume–energy relationship.
a¼
C 11 þ C 12 2C 13 . C 33 C 13
(9) (10)
Table 1 The calculated structural parameters compared with experiments and other theoretical results a (A˚)
c (A˚)
c/a
B0 (GPa)
B0
B00 (GPa1)
Present work
3.093
3.520
1.138
3.63a 3.41b
0.034 0.029
Vogt et al. [6] Singh [7] Islam et al. [8] Osorio-Guillen et al. [9] Ravindran et al. [10] Experiments [1,6]
3.089 3.048 3.064 3.085 3.080 3.086
3.548 3.487 3.493 3.557 3.532 3.521
1.149 1.144 1.140 1.153 1.147 1.141
154.31a 153.61b 150c 139710
a
Obtained from the natural strain EOS [28]. Obtained from the quasi-harmonic Debye model [22]. c Obtained from Eq. (11). b
122 151
3.4 3.65
15175
4.0
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Applying the calculated elastic constants at zero pressure, i.e. C 11 ¼ 462, C 12 ¼ 67, C 13 ¼ 41, C 33 ¼ 254, and C 44 ¼ 80 GPa [29], we obtain the bulk modulus Ba and Bc are 615.04, 293.07 GPa, respectively, that is, the compression along the aaxis is more difficult than that along the c-axis. This phenomenon can be understood by the knowledge of the bonding situations in MgB2, which is characterized as a strong covalent bonding in the boron layers, ionized magnesium atoms and a weaker interlayer metallic bonding. These mechanical properties suggest the anisotropy of MgB2. From the following formula of the zero-pressure bulk modulus B0 : B0 ¼
C 33 ðC 11 þ C 12 Þ 2C 213 , C 11 þ C 12 4C 13 þ 2C 33
(11)
we can obtain B0 ¼ 150 GPa. In the case, there is no constraint on the c=a dependence on lattice strain [30]. The calculated result agrees excellently with the experimental data [6]. In Fig. 2, we illustrate the pressure P dependence of the normalized volume V n ð¼ V =V 0 Þ of MgB2 in the range of 20–50 GPa, together with the experimental data at P ¼ 0240 GPa [31] and the theoretical results by Islam et al. [8]. The results obtained in this work consist with the experimental data [31], and seem to be better than
Fig. 3. Pressure dependence of the normalized lattice parameters and c=a ratio of MgB2.
those by Islam et al. [8]. The normalized lattice parameters a=a0 , c=c0 and the ratio c=a as a function of pressure are plotted in Fig. 3, where a0 and c0 are their values at T ¼ 0 and P ¼ 0, respectively. By fitting the calculated data to second-order polynomials, we obtain the following relationships at T ¼ 0 K : a=a0 ¼ 1:000 0:00190P þ 1:46159 105 P2 , (12) c=c0 ¼ 1:000 0:00342P þ 2:65743 105 P2 , (13) c=a ¼ 1:138 0:00172P þ 1:20368 105 P2 . (14)
Fig. 2. The normalized volume V n as a function of pressure P. The solid lines with square, circular and trigonal symbols represent the present work, experimental data and the other theoretical results obtained by Islam et al. [8], respectively.
These results agree with those obtained by Jorgensen et al. [32] through the neutron diffraction measurements under low pressure (o 0.62 GPa), also agree with those measured using a Merrill–Bassett diamond anvil cell up to 8 GPa [6]. It is shown that, as pressure increase, the equilibrium ratio c=a ratio decrease. The compression along the c-axis is much larger than that along the a-axis, consistent with the comparatively weaker (Mg–B) bonds that determine the c-axis length. The obtained isothermal V n 2P and isobaric V 2T relationship curves in the wide range of 0–240 GPa are illustrated in Figs. 4 and 5,
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respectively. For the isothermal compressions shown in Fig. 4, the volume is greatly affected by both the pressure and the temperature. The effect of increasing pressure on MgB2 is the same as decreasing temperature of MgB2. On the other hand, as the pressure increases, the volume compression decreases at a given temperature, and the volume compression at high temperature is less than that at lower temperature under a given pressure. As for the isobaric curves shown in Figs. 5(a) and (b), we find that, under lower pressure, the volume varies quickly as the temperature rise. Under higher pressure, it becomes moderate, and the V 2T relations are nearly linear. The V 2T relation at P ¼ 0 GPa (Fig. 5(a)) in the range of temperatures 0–300 K is consistent with the neutron powder diffraction measurements [32]. The compression behaviors of MgB2 correspond to the bonding situations in MgB2. When pressure increases, the atoms in the interlayers become closer, and the interactions between these atoms become stronger. Compared to K3C60, a fairly soft material with d ln V =dP ¼ 0:036 GPa1 [33], MgB2 is a tightly packed incompressible solid with d ln V =dP ¼ 0:0076 GPa1 . MgB2 is less compressible in the basal plane, in which the covalent B–B bonds lie. The interlayer linear compressibility (d ln c=dP ¼ 0:00353 GPa1 ) is about 1.8 times larger than that the in-plane value (d ln a=dP ¼ 0:00196 GPa1 ).
Fig. 5. The volume–temperature relationship curves: (a) under lower pressures, and (b) under higher pressures.
Fig. 4. The volume compressions V n of MgB2 up to pressure 240 GPa at the temperatures of 300, 1000, and 2000 K, respectively.
In summary, we apply the full-potential linearized muffin-tin orbital (FP-LMTO) scheme within the generalized gradient approximation to calculate the structure and equation of state of MgB2 at high pressure. It is demonstrated that the ratio c=a of about 1.138 is the most stable structure for hcp MgB2, the calculated zero pressure bulk module B0 and the pressure derivation of bulk modulus B00 from the natural strain EOS and from the quasiharmonic Debye model are about 154.31 GPa and 3.63, 153.61 GPa and 3.41, respectively. Through the quasi-harmonic Debye model, the dependences of the volume compression on pressure and the volume on temperature are successfully obtained. The authors would like to thank the support by the National Natural Science Foundation of
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