First-principles calculations of thermodynamic properties of TiB2 at high pressure

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Physica B 400 (2007) 83–87 www.elsevier.com/locate/physb

First-principles calculations of thermodynamic properties of TiB2 at high pressure Feng Penga,, Hong-Zhi Fua,b, Xin-Lu Chengb a

College of Physics and Electronic Information, Luoyang Normal College, Luoyang 471022, People’s Republic of China b Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, People’s Republic of China Received 17 June 2007; accepted 22 June 2007

Abstract The equations of state (EOS) and other thermodynamic properties of TiB2 are investigated using ab initio plane-wave pseudopotential density functional theory method. The obtained results are in good agreement with the experimental measured data and other theoretical calculated ones. Through the quasi-harmonic Debye model, in which the phononic effects are considered, the dependences of relative volume V/V0 on pressure P, cell volume V on temperature T, and Debye temperature Y and specific heat CV on pressure P are successfully obtained. r 2007 Elsevier B.V. All rights reserved. PACS: 64.30.+t; 51.30.+I; 71.15.Mb; 91.60.x Keywords: Equations of state; Thermodynamic properties; Generalized gradient approximation; TiB2

1. Introduction Titanium diboride is one of the well-known materials which presents a very attractive combination of mechanical, chemical and transport properties such as high elastic modulus (about 565 GPa), high hardness (about 24 GPa), high melting point (about 3500 K), high specific strength, low density (4495 kg m3), low electrical resistivity (0.13 mOm), good thermal conductivity (ranging from 37 to 122 Wm1 K1, at 300 K) and excellent chemical inertness [1–5]. These characteristics make it a potential candidate for several high-performance applications, in cutting tools, crucibles, electrodes and wear-resistant components [1–3]. Titanium diboride crystallizes in the hexagonal AlB2 structure with the space group P6/mmm. There are three atoms in the unit cell, all of them on the special positions: the titanium atom at the origin and two boron atoms at the site 2d (1/3, 2/3, 1/2). The structure is thus extremely Corresponding author. Tel.: +86 379 62960015; fax: +86 379 65526093 E-mail address: [email protected] (F. Peng).

0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.06.020

simple in that it is defined by two lattice parameters, a and c, and has a very small unit cell. Experimental estimates of the cell parameters extrapolated to 0 K give a ¼ 3.0236 A˚ and c ¼ 3.2204 A˚ [6]. A hexagonal TiB2 crystal has six different elastic coefficients (C11, C12, C13, C33, C44 and C66), but only five of them are independent since C66 ¼ 12 (C11 C12). Despite the prominence that TiB2 has found in hightechnology applications, some of its properties are scarcely known. Such as, to our knowledge there is a few published result [7] for the pressure dependence of the TiB2 lattice parameters, unit cell volume and thermodynamic properties. There are several theoretical studies in the literature, involving different approaches, where main concern is the electronic structure of the transition-metal diborides, in particular those with the AlB2 structure [7–14]. However, only few of them concern the pressure behavior, elastic and thermodynamic properties of these compounds. But, the study of the pressure behavior of very hard materials is necessary. So, in this work, we investigate the equations of state (EOS) and the other thermodynamic properties of TiB2 in the range of 0–240 GPa by the plane-wave pseudopotential density functional theory method through

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the Cambridge Serial Total Energy Package (CASTEP) program [15,16] and by the quasi-harmonic Debye model [17], through which all thermodynamic quantities are obtained from the calculated energy–volume points. The lattice constant, the bulk modulus B0, the pressure derivative of bulk modulus, elastic constants and the other thermodynamic properties of TiB2 are obtained successfully. 2. Theoretical method In the electronic structure calculations, we use the normconserving pseudopotential in Kleinman–Bylander representation generated using the optimization scheme of Lin et al. [18] in order to reduce the required value of the planewave energy cut-off. The generalized gradient approximation (GGA) exchange-correlation function [19] is represented by revised Perdew Burke Ernzerhof (RPBE) formula. A plane-wave basis set with energy cut-off 400.00 eV is applied. Pseudo atomic calculations are performed for Ti 3d24s2 and B 2s22p1. For the Brillouinzone sampling, we use the 10  10  8 Monkhorst–Pack mesh, where the self-consistent convergence of the total energy is at 5.0  106 eV/atom. All the total energy electronic structure calculations are implemented through the CASTEP code [15,16]. To obtain the thermodynamic properties of TiB2, the quasi-harmonic Debye model [17] is applied, in which the non-equilibrium Gibbs function G*(V;P,T) takes the form [17] G  ðV ; P; TÞ ¼ EðV Þ þ PV þ Avib ðYðV Þ; TÞ,

(1)

where Y(V) is the Debye temperature, and the vibrational term Avib can be written as [20,21]   9Y Y=T þ 3 lnð1  e Avib ðY; TÞ ¼ nKT Þ  DðY; TÞ , (2) 8T where D(Y/T) represents the Debye integral, n is the number of atoms per formula unit. For an isotropic solid, Y is expressed by [20]: rffiffiffiffiffiffi _ h 2 1=2 i1=3 Bs 6p V n Y¼ f ðsÞ , (3) K M where M is the molecular mass per formula unit, Bs the adiabatic bulk modulus, which can be approximated by the static compressibility [17]  Bs ffi BðV Þ ¼ V

 d2 EðV Þ , dV 2

(4)

the Poisson ratio s is taken as 0.114 [22], f(s) is given by Refs. [23,24]. Therefore, the non-equilibrium Gibbs function G*(V;P,T) as a function of (V; P, T) can be minimized concerning volume V    @G ðV ; P; TÞ ¼ 0. @V P;T

(5)

The thermal EOS is obtained by solving Eq. (5). The isothermal bulk modulus BT, the heat capacity CV are expressed as    @G ðV ; P; TÞ , @V P;T

(6)

 3Y=T C V ¼ 3nk 4DðY=TÞ  Y=T . e 1

(7)

BT ðP; TÞ ¼ V 

3. Results and discussion To obtain the total energy E and the corresponding volume V for TiB2, we take a series of different lattice parameters a and c. The calculated E–V points are then fitted to the natural strain EOS [25], in which the pressure–volume relationship expanded to the fourth order in strain is    V0 3 P ¼ 3B0 f N 1 þ ðB0  2Þf N 2 V  3 þ ð1 þ B0 B þ ðB0  2Þ þ ðB0  2Þ2 Þf 2N , ð8Þ 2 where f N ¼ lnðl=l 0 Þ, which may be written as f N ¼ 1 3 lnðV =V 0 Þfor 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  1  (9) 1 þ ðB0  2Þ þ ðB0  2Þ2 . B00 ¼ B0 The obtained lattice constants a and c, bulk modulus B0 and pressure derivative of bulk modulus B00 for TiB2 at zero pressure are listed in Table 1. Obviously, the calculated results are well consistent with the other theoretical results [7,14,26–28] and the experimental data [6,29,30]. Moreover, we calculated all the elastic constants, bulk modulus B and the shear modulus A at zero pressure as listed in Table 2, which agree well with the measured data [22,31–33] and the other calculated results [26,27,34]. In order to provide some insight into the pressure behavior of TiB2, the total energy of titanium diboride was minimized as a function of the primitive cell volume as plotted in Fig. 1. Fig. 2 shows the dependence of the volume of the TiB2 on pressure. Clearly, the present results are in good agreement with the experimental data [35] and the other theoretical result [26]. Table 3 shows the present calculated elastic constants depending on the pressure. It is found that C12 and C13 vary a little largely under the effect of pressure as compared with the variations in C11, C33, C44 and C66. The reason is that C12 and C13 are sensitive to pressure as compared with C11, C33, C44 and C66. But the sensitivity of the effect of pressure is not obvious, which implies that the anisotropy of TiB2 is very weak. The two anisotropies of compressibility (Acomp) and shear (Ashear) indices were calculated and

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Table 1 The lattice constants, c/a ratio, volume, aggregate elastic moduli (GPa) and pressure derivative of bulk modulus for TiB2 at zero pressure

Theoretical Present work van Camp and van Doren [14]a van Camp and van Doren [14]b Perottoni et al. [26] Milman and Warren [27]c Milman and Warren [27]d Tian and Wang [7] Vajeeson et al. [28] Experimental Munroe [6]

Silver and Kushida [29] Post et al. [30]

a (A˚)

C(A˚)

c/a

V (A˚3)

B0 (GPa)

B00

3.015 3.023 2.993 3.027 3.029 2.991 2.895 3.070

3.222 3.166 3.147 3.240 3.219 3.152 3.086 3.262

1.068 1.047 1.051 1.070 1.063 1.054 1.066 1.063

25.4 25.1 24.4 25.7

245 270 260 292 250.6(1) 277.2(3) 370 213

3.88

3.0236

3.2204

1.065

25.5

237(16) 205(11) 247(12)

2.0(2) 2.4(2) 1.9(3)

3.028 3.030

3.228 3.230

1.066 1.066

25.6 25.7

3.34 3.86 3.84 2.1

a

Density functional theory (DFT) with ab initio norm-conserving pseudopotentials in the local density approximation (LDA). The same as above, but in the generalized gradient approximation (GGA). c DFT with ab initio ultrasoft pseudopotentials in GGA. d The same as above, but in the LDA. b

Table 2 Elastic constants (GPa) of TiB2 at zero pressure

Theoretical Present work Panda and Ravi Chandran [34]a Panda and Ravi Chandran [34]b Milman and Warren [27]c Milman and Warren [27]d Perottoni et al. [26] Experimental Spoor et al. [31] Gilman and Roberts [32] Manghnani et al. [33] Wright [22]

C11

C12

C13

C33

C44

C66

B

A

626 654 650 659 655 786

68 75 79 62 65 127

102 99 100 100 99 87

444 443 443 461 461 583

240 344 256 260 260 271

279 290 285 299 295 329

245 249 249 251 251 292

0.860 1.186 0.898 0.869 0.881 0.824

660 690 588 672

48 410 72 40

93 320 84 125

432 440 503 224

260 250 238 232

306 140 258 316

240 399 239 194

0.850 1.786 0.922 0.734

a

Density functional theory (DFT) with full potential linearized augmented plane wave (FLAPW), no relaxation of internal degrees of freedom. DFT with FLAPW, relaxation of internal degrees of freedom. c DFT with ultrasoft pseudopotentials in generalized gradient approximation, no relaxation of internal degrees of freedom. d The same as above, eight strain patterns up to the maximum amplitude of 2%. b

are listed in Table 4. The ‘percentage anisotropies’ for TiB2, as derived from our set of calculated elastic constants in Table 2, are given in Table 4, where they are compared to the values calculated from the single-crystal elastic constants measured by Gilman and Roberts [32], Spoor et al. [31] and Perottoni et al. [26]. As can be seen, a fair agreement is observed with the results of Spoor et al, supporting their conclusion that TiB2 is more isotropic than previously supposed. Fig. 3 illustrates the relative volumes of TiB2 at various pressures and temperatures; from this isothermal curves it can be seen that, as the pressure P increases, the relative volume V/V0 decreases at a given temperature, and the relative volume V/V0 of higher temperature is less than that of lower temperature at the same pressure. On the other

hand, volume V decreases with the elevated pressure P, and increases with the elevated temperature T. This effect of increasing pressure on TiB2 is just the same as decreasing temperature of TiB2. The Cv and the Debye temperature Y as a function of pressure P are shown in Fig. 4 at the temperatures of 300 and 1500 K for TiB2. It is shown that when the temperature is constant, the debye temperature Y increases almost linearly with applied pressures. However, the Cv decreases with the applied pressures, as is due to the fact that the effect of increasing pressure on TiB2 is the same as decreasing temperature of TiB2. As for the isobaric curves shown in Figs. 5 and 6, we find that, under lower pressure, the volume varies quickly as the temperature increases. Under higher pressure, it becomes

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Table 4 Elastic anisotropy of TiB2 Percentage anisotropy

Energy(Hartree)

-8.6

-8.8

Present work Spoor et al. [31] Panda and Ravi Chandran [34] Perottoni et al. [26] Gilman and Roberts [32]

25.4

-9.0

Acomp

Ashear

Acomp ð%Þ

Ashear ð%Þ

1.43 1.54 1.146

0.86 0.85 0.896

1.26 1.27 1.25

1.44 1.54 1.0946

3.83

1.79

1.3 4.40

0.7 6.32

-9.2

-9.4 10

15

20

25

30

35

1.05

Primitive cell volume(angstrom3)

1.00

Fig. 1. Energy–volume relationship.

300K 1000K 1500K

0.95

V/V0

0.90

1.00 Present work Perottoni et al. Exp.

0.95 0.90

0.85 0.80 0.75

V/V0

0.70

0.85

0.65

0.80

0

50

0.75

100 Pressure(GPa)

150

200

Fig. 3. The relative volume versus pressure of TiB2 at temperature of 300, 1000 and 1500 K, respectively.

0.70 0.65 0

50

100 Pressure(GPa)

150

200

Fig. 2. The V/V0–pressure relationship.

Θ 300K Θ 1500K CV 300K CV 1500K

P

C11

C12

C13

C33

C44

C66

B

A

0 10 20 30 40 50 60 70 80 90 100 110 120

626 695 755 821 874 930 984 1037 1086 1122 1165 1218 1253

68 85 103 122 137 155 176 195 212 250 269 289 302

102 131 161 190 218 245 272 297 318 340 364 391 417

444 494 541 593 670 667 718 758 807 850 889 931 951

240 268 299 325 353 377 401 425 447 463 484 506 523

279 305 326 350 369 388 404 421 437 436 448 465 475

245 282 318 355 387 419 453 484 515 546 575 607 632

0.860 0.879 0.917 0.929 0.957 0.972 0.993 1.009 1.023 1.062 1.080 1.088 1.101

(X-X0)/X0

0.5 Table 3 Elastic constants (GPa) of TiB2 at various pressures P (GPa) and at 300 K

0.0

-0.5 0

50

100 Pressure(GPa)

150

200

Fig. 4. Variations of thermodynamic parameters (Debye temperature or specific heat) with pressure P. They are normalized by (X–X0)/X0 where X and X0 are the Debye temperature or heat capacity under any pressure P and zero pressure P0 at the temperatures of 300 and 1500 K.

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References

190 0GPa 10GPa 20GPa

Volume(Bohr3)

185 180 175 170 165 160 0

500

1000 1500 Temperature(K)

2000

2500

Fig. 5. The volume–temperature relationship curve at lower pressure.

119

200GPa 220GPa 240GPa

118 Volume(Bohr3)

87

117 116 115 114 113 112 0

500

1000 1500 Temperature(K)

2000

2500

Fig. 6. The volume–temperature relationship curve at higher pressure.

moderate, and the V–T relations are nearly linear. The compression behaviors of TiB2 correspond to the bonding situations in TiB2. As pressure increases, the atoms in the interlayers become closer, and the interactions between these atoms become stronger. Acknowledgments The authors would like to thank the support by the National Natural Science Foundation of China under Grant nos. 10376021, 10274055, and by the Research Fund for the Doctoral Program of High Education of China under Grant no. 20020610001.

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