First-principles calculations of vibrational and thermodynamical properties of rare-earth diborides

Computational Materials Science 68 (2013) 307–313

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First-principles calculations of vibrational and thermodynamical properties of rare-earth diborides Haci Ozisik a,⇑, Kemal Colakoglu b, Engin Deligoz c, Engin Ateser c a

Aksaray University, Education Faculty, Department of Computer and Instructional Technologies Teaching, 68100 Aksaray, Turkey Gazi University, Department of Physics, Teknikokullar, 06500 Ankara, Turkey c Aksaray University, Department of Physics, 68100 Aksaray, Turkey b

a r t i c l e

i n f o

Article history: Received 23 March 2012 Received in revised form 30 October 2012 Accepted 1 November 2012

Keywords: Ab initio Rare earth diborides Dynamical properties Thermodynamic properties

a b s t r a c t We have tried to theoretically predict the lattice dynamical and thermodynamic properties of rare earth diborides (XB2, X = Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm) in AlB2-type structure based on density functional theory within generalized gradient approximation. The calculated equilibrium lattice parameters are in overall agreement with the available experiment and other theoretical results. The phonon dispersion curves are derived by using the direct method. The lattice dynamical results are showed that these compounds are dynamically stable for the considered structure, and these properties exhibits, almost, similar trend for these compounds. In addition, thermodynamic properties such as the free energy, enthalpy, entropy and heat capacity are, also successfully predicted and analyzed with the help of phonon dispersion. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Hard materials are of considerable fundamental interest and practical importance due to their numerous technological applications, from cutting and polishing tools to wear-resistant coatings [1]. Unfortunately, almost all super hard materials (diamond, cubic BN, etc.) are expensive because they either occur naturally in limited supplies or have to be made at high pressure synthetically [2]. A promising approach to design super hard materials is to combine transition metals possessing a high bulk modulus with small, covalent bond forming atoms such as boron, carbon, nitrogen or oxygen [2–4]. Therefore, intense research efforts have been carried out to design the new super hard materials such as OsB2 [5], ReB2 [4,5], LuB2 [6], PtN [7], IrN2 [8], Re2C [9,10], and Ta2N3 [11,12] using the light elements. A few experimental and theoretical studies of rare-earth diborides have been reported [6,13–27] so far. Recently, we [6] have calculated mechanical and dynamical properties of LuB2 compound; according to our results it is both mechanically and dynamical stable. Zazoua et al. [25] have studied the structures and magnetic phase stability of rare earth diboride compounds, XB2, using density functional simulations within the local density approximation. Their results showed that these compounds are incompressible materials. The elastic constants of XB2 compounds in AlB2-type structure have calculated by Duan et al. [26] using

⇑ Corresponding author. Tel.: +90 382 2882258; fax: +90 382 2882226. E-mail address: [email protected] (H. Ozisik). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2012.11.003

first-principles with the generalized gradient approximation. They have concluded that these compounds are all mechanically stable and AlB2-type compounds with high anisotropy have ductile. They have concluded that these compounds are all mechanically stable in AlB2-type structure where the compounds are ductile with high anisotropy. Kacimi et al. [27] have studied the electronic structure and magnetic behaviors of hexagonal rare-earth diborides RB2 are using ab initio density functional theory. They have suggested that RB2 (R = Pr, Gd, Tb, Dy, Ho, Er, and Tm) compounds are ferromagnetic. Recently, we have investigated structural and mechanical properties of rear earth diborides (Sm–Tm) compounds [28]. A number of studies [29–37] exist in the literature dealing with lattice dynamical properties of borides and diborides. The results of these works showed that diborides and borides are dynamically stable for AlB2 and WC structures, respectively. But up to now – to our best knowledge –, the vibrational and thermodynamical properties, which are the important bulk properties of a solid, have not been studied theoretically or experimental for XB2 (X = Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm) compounds in AlB2-type structure (space group P6/mmm). Systemic studies on lattice dynamics and thermodynamics of rare earth borides are of great importance and in demand. The phonon dispersion curve provides important information about the dynamical properties of materials. Many physical properties of solids depend on their phonon properties, such as specific heat, thermal expansion, free energy, heat conduction, sound velocity and electron–phonon coupling. In addition, low frequency modes can be associated with phase transformations, while imaginary frequencies show that the calculated structure is not the most stable

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one. Consequently, the main purpose of this work is to fill this gap by investigating dynamical and thermodynamical properties of these compounds by using the ab initio total energy calculations. The calculation method is given in Section 2; the results are discussed in Section 3. Finally, the summary and conclusion are presented in Section 4. 2. Methodology 2.1. Structural properties In the present paper, all calculations have been carried out using the Vienna ab initio simulation package (VASP) [38–41] based on the density functional theory (DFT). The electron–ion interaction was considered in the form of the projector-augmented-wave (PAW) method with plane wave up to energy of 450 eV [42]. This cut-off was found to be adequate for studying the lattice dynamical properties. For the exchange and correlation terms in the electron–electron interaction, Perdew–Burke–Ernzerhof (PBE) type functional [43] was used within the generalized gradient approximation (GGA). For k-space summation the 11  11  11 Monkhorst and Pack [44] gamma centered grid of k-points have been used. The standard PAW pseudopotential for B element was used. In the PAW method, the core electrons are replaced by an effective potential and only the remaining electrons is treated explicitly. For Sm–Lu elements, we have employed the frozen-core rare-earth

pseudopotentials in which the valencies are set to 3 and the remaining f electrons spatially localized near the core are treated as core-electrons. Actually, in the Vasp package [38–41], special GGA potentials are supplied for Ce–Lu, in which f-electrons are kept frozen in the core (standard model for the treatment of localized f electrons). The number of f-electrons in the core equals the total number of valence electrons minus the formal valency. For instance: according to the periodic table Sm has a total of 8 valence electrons (6f electrons and 2s electrons). These treatments improve the quality of pseudopotentials and can almost reproduce the results of accurate all-electron calculations that do not include a frozen core; then one can obtain the more reliable results for materials with strong magnetic moments or with atoms that have large differences in electronegativity [45]. For XB2 within the AlB2-type structure (Space group: No. 191, P6/mmm), the primitive cell contains one X (rare-earth atom) and two Boron atoms occupy in nonequivalent atomic positions of 1a (0, 0, 0) and 2d (1/3, 2/3, 1/2), respectively. 2.2. Lattice dynamical and thermodynamic properties The present phonon dispersion curves and phonon density of states are calculated using the direct method [46] as implemented in the Phonon Software [47] in a similar manner to our recent works [7,38,39]. The force constants and dynamical matrix are obtained from the Hellmann–Feynman forces evaluated with small individual displacements of nonequivalent atoms. The displacement

Table 1 The equilibrium lattice parameters (a and c in Å) and c/a ratios for XB2 compounds in AlB2-type along with the available experimental and theoretical values. Comp.

a

C

c/a

Refs.

Comp.

a

c

c/a

Refs.

CeB2

3.362 3.152 3.180 3.348 3.092 3.160 3.338 3.108 3.202 3.33 3.142 3.195 3.327 3.31 3.31 3.158 3.211 3.322 3.324 3.198 3.188 3.317 3.315 3.319 3.236 3.245 3.310 3.318 3.293 3.309 3.28 3.29 3.2986 3.294 3.285 3.184 3.231 3.284

4.253 4.113 4.112 4.190 4.080 3.969 4.135 4.093 4.039 4.077 4.089 4.069 4.036 4.019 4.019 4.095 4.092 4.024 3.980 4.056 4.087 3.939 3.936 3.944 3.883 3.892 3.940 3.933 3.947 3.901 3.86 3.878 3.879 3.886 3.892 3.811 3.800 3.912

1.265 1.305 1.293 1.251 1.320 1.256 1.239 1.317 1.261 1.224 1.301 1.274 1.213 1.205 1.214 1.296 1.274 1.211 1.197 1.268 1.281 1.188 1.187 1.188 1.199 1.199 1.190 1.185 1.199 1.179 1.177 1.179 1.176 1.180 1.1848 1.197 1.176 1.191

Present (LSDA) Ref. [25] (LSDA + U) Ref. [25] Present (LSDA) Ref. [25] (LSDA + U) Ref. [25] Present (LSDA) Ref. [25] (LSDA + U) Ref. [25] Present (LSDA) Ref. [25] (LSDA + U) Ref. [25] (GGA) Ref. [28] (LDA) Ref. [14] (Exp.) Ref. [15] (LSDA) Ref. [25] (LSDA + U) Ref. [25] (GGA) Ref. [26] (GGA) Ref. [28] (LSDA) Ref. [25] (LSDA + U) Ref. [25] (GGA) Ref. [28] (Exp.) Ref. [15] (Exp.) Ref. [19] (LSDA) Ref. [25] (LSDA + U) Ref. [25] Ref. [24] Ref. [26] (GGA) Ref. [26] (GGA) Ref. [28] (LDA) Ref. [14] (Exp.) Ref. [16] (Exp.) Ref. [13] (Exp.) Ref. [13] (Exp.) Ref. [21] (LSDA) Ref. [24] (LSDA + U) Ref. [25] (GGA) Ref. [26]

DyB2

3.298 3.287 3.286 3.292 3.290 3.179 3.213 3.286 3.273 3.279 3.257 3.281 3.178 3.212 3.170 3.281 3.275 3.28 3.2681 3.271 3.2683 3.265 3.171 3.202 3.268 3.262 3.25 3.258 3.2609 3.261 3.2548 3.26 3.257 3.261 3.171 3.153 3.254 3.242 3.242

3.865 3.847 3.852 3.852 3.847 3.792 3.822 3.832 3.814 3.811 3.759 3.816 3.790 3.756 3.810 3.811 3.801 3.79 3.783 3.782 3.781 3.770 3.781 3.752 3.785 3.769 3.739 3.745 3.753 3.755 3.755 3.754 3.747 3.755 3.767 3.850 3.761 3.717 3.73

1.172 1.170 1.172 1.170 1.169 1.193 1.189 1.166 1.165 1.162 1.154 1.163 1.193 1.169 1.202 1.162 1.161 1.156 1.158 1.156 1.157 1.155 1.192 1.172 1.158 1.155 1.151 1.150 1.151 1.151 1.154 1.151 1.150 1.152 1.188 1.221 1.156 1.146 1.150

(GGA) Ref. [28] (LDA) Ref. [14] (Exp.) Ref. [21] (Exp.) Ref. [18] (Exp.) Ref. [18] (LSDA) Ref. [25] (LSDA + U) Ref. [25] (GGA) Ref. [28] (LDA) Ref. [14] (Exp.) Ref. [15] (Exp.) Ref. [21] (Exp.) Ref. [20] (LSDA) Ref. [25] (LSDA + U) Ref. [25] Ref. [24] Ref. [25] (GGA) Ref. [28] (LDA) Ref. [14] (Exp.) Ref. [13] (Exp.) Ref. [13] (Exp.) Ref. [21] (Exp.) Ref. [20] (LSDA) Ref. [25] (LSDA + U) Ref. [25] (GGA) Ref. [26] (GGA) Ref. [28] (LDA) Ref. [14] (Exp.) Ref. [15] (Exp.) Ref. [13] (Exp.) Ref. [13] (Exp.) Ref. [21] (Exp.) Ref. [22] (Exp.) Ref. [23] (Exp.) Ref. [24] (LSDA) Ref. [25] (LSDA + U) Ref. [25] (GGA) Ref. [26] (GGA) Ref. [6] (GGA) Ref. [26]

PrB2

NdB2

PmB2

SmB2

EuB2

GdB2

TbB2

HoB2

ErB2

TmB2

LuB2

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700

CeB 2

PrB 2

Ce B

Pr B

NdB 2

Frequency (cm-1)

600 500 400 300 200

Nd B

100 0 Γ 700

K M

Γ A

H L

Γ

A

K M

Γ A

H L

Γ

A

K M

Γ A

H L

A

PmB 2

SmB 2

EuB 2

Pm B

Sm B

Eu B

Frequency (cm-1)

600 500 400 300 200 100 0 Γ 700

K M

Γ

A

H L

Γ

A

K M

Γ A

H L

Γ

A

GdB 2

TbB 2

Gd B

Tb B

K M

Γ

A

H L

A DyB 2

Frequency (cm-1)

600 500 400 300

Dy B

200 100 0 Γ 700

K M

Γ

A

H L

Γ

A

K M

Γ

A

H L

Γ

A

HoB 2

ErB 2

Ho B

Er B

K M

Γ

A

H L

A TmB 2

600

Frequency (cm-1)

500 400 300

Tm B

200 100 0

Γ

K M

Γ

A

H L

A

Γ

K M

Γ

A

H L

A

Γ

K M

Γ

A

H L

A

Fig. 1. Calculated phonon dispersions and partial density of states of XB2 compounds in the AlB2-type structure.

amplitudes are taken as 0.03 Å, to increase the precision and minimize the anharmonic effects, and the positive and negative atomic displacements along x, y, and z axes are taken into account.

As is known, the thermodynamic properties of the crystals can be determined in detail by phonons. Traditionally, the Gibbs free energy G (T, P) is defined as

H. Ozisik et al. / Computational Materials Science 68 (2013) 307–313

GðT; PÞ ¼ FðT; VÞ þ PV;

165

where F(T, V) is the free energy of the crystal and it is sum of the ground state energy (E(V)) and the phonon free energy (Fph(T, V)) at a given unit cell volume V as follows:

160

FðT; VÞ ¼ EðVÞ þ F ph ðT; VÞ:

ð2Þ

The temperature T appears in Fph(T, V) via the phonon term. The more explicit forms of the internal energy, free energy, entropy, and heat capacity in the harmonic approximation per primitive unit cell are given [47] as follows:



140 135 130

ð3Þ

      hx hx hx gðxÞ cothð Þ  1  ln 1  exp dx; 2kB T 2kB T 2kB T ð5Þ

 exp khBxT h  i2 dx; exp khBxT  1

ð6Þ

2

2

Tm B 2 Lu B

2

Sm B 2 Eu B 2 G dB 2 Tb B 2 D yB 2 H oB 2 Er B

B

660 640 620

2

B Lu

B Tm

2

2

2

600 Sm B 2 Eu B 2 G dB 2 Tb B 2 D yB 2 H oB 2 Er B

where r is the number of degree of freedom in the unit cell,  h is the Planck constant, kB is the Boltzmann constant, T is the temperature, x denotes the phonon frequency, and g(x) is the phonon density of states.

2

2

B

x h gðxÞ kB T

Pm



2

0

1

680

eB

Z

700

C

1

Pm

(b)

720

0

C ¼ rkB

740

Phonon Max [cm-1]

S ¼ rkB Z 

ð4Þ

2

0

   hx dx; gðxÞ ln 2 sinh 2kB T

B

1

C

Z

Pr

F ¼ rkB T

2

125 2

0

 x h dx; 2kB T

145

dB

gðxÞðhxÞ cot h

150

N

1

155

dB

Z

1 r 2

(a)

eB 2 Pr B

E¼

Phonon Gap [cm-1]

ð1Þ

N

310

Fig. 2. (a) Calculated gap between X–B contributions (in cm1) and (b) maximum LO frequencies (in cm1) of XB2 compounds in the AlB2-type structure.

3. Results and discussion 3.1. Lattice dynamical properties We have fully optimized the lattice parameters for XB2 (X = Ce– Pm) and the optimized the lattice parameters are taken from our recent study for XB2 (X = Sm–Lu) compounds [6,28] to obtain lattice dynamical properties. The lattice parameters are listed in Table 1 along with the available other works for comparison. It is seen that the present lattice parameters (a, c) are in good agreement with the other works [13–26]. The obtained phonon dispersion curves along the high-symmetry directions and the corresponding phonon DOS for these compounds are illustrated in Fig. 1. The overall features of the phonon dispersion curves are similar to those of the other transi-

tion-metal and rare earth borides [6,32,33]. The unit cell of AlB2type structure contains three atoms, which give rise to a total of nine phonon branches, which contains three acoustic modes and six optical modes. The present phonon dispersion curves do not contain soft modes at any wave vectors, and there are no optical phonon branches with dispersions that dip toward the zero frequency. This confirms that the considered compounds are dynamically stable in AlB2-type structure. All phonon dispersion curves presented in Fig. 1 show a clear band gap between the low and high frequency region, and phonon density of states curves are divided by the gaps, the same to the phonon dispersion curves. The calculated phonon band gaps and maximum value of the phonon frequencies are given in Table 2

Table 2 Calculated valance electron concentration (electron/atom) free energies at 0 K (in meV/u.c.), phonon gap between X–B contributions (in cm1), maximum LO frequencies (cm1), Entropy (in kB/u.c.), and heat capacity (in kB/u.c.) for RE-diborides in AlB2-structure.

a

Materials

VEC

Free energy

Phgap

LOmax

Entropy (300 K)

Entropy (1000 K)

Cv (300 K)

Cv (1000 K)

CeB2 PrB2 NdB2 PmB2 SmB2 EuB2 GdB2 TbB2 DyB2 HoB2 ErB2 TmB2 LuB2a

6.00 6.33 6.67 7.00 7.33 7.67 8.00 8.33 8.67 9.00 9.33 9.67 10.33

185.9 189.2 191.9 194.4 196.3 200.2 203.6 207.9 213.1 217.7 222.0 226.5 232.8

126 129 128 129 132 130 134 131 134 140 148 155 163

606 615 624 631 636 642 649 655 665 674 685 699 727

7.802 7.665 7.504 7.378 7.009 7.041 6.665 6.755 6.601 6.465 6.343 6.218 6.083

17.662 17.495 17.312 17.164 16.744 16.781 16.296 16.416 16.219 16.037 15.871 15.701 15.493

7.070 7.016 6.977 6.940 6.846 6.859 6.672 6.735 6.649 6.573 6.502 6.429 6.326

8.785 8.778 8.772 8.767 8.755 8.756 8.728 8.738 8.725 8.713 8.701 8.689 8.671

The results are taken from our previous study [6].

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and shown in Fig. 2a and b. It can be seen that the band gap increases almost linearly from TbB2 to LuB2. We note that the anion atoms are the same in these compounds, the cation atom’s size are different. This difference could be the responsible for increasing of the maximum value of the phonon frequencies from CeB2 to LuB2 (see Fig. 2b). For these compounds, interesting features of optical phonon modes are observed at C point. Optical phonon branches are nearly flat at the C point and the magnitude of acoustic phonon modes is less than the optical ones. E2g mode evolves from the doubly degenerate at the C–A directions for this structure; but the same branches are not degenerate along the A–L and C–K directions. The energy of this mode also increases when the size of the cation mass increases (Ce ? Lu). The acoustic branches have steep slopes indicating large elastic coefficients. On the right side of phonon curves, the corresponding partial density of phonon states is also plotted for the considered compounds. While the low-frequency phonon modes are dominated by motion of the X atoms, the high-frequency phonon modes are mainly characterized by motion of the B atoms. This is expected because the boron atom is lighter than rare-earth atoms and strong B– B coupling, which leads to comparatively weaker electron–phonon interactions. The covalent character of the B–B bonding is also decisive for the high frequency of phonons involving the boron atoms. 3.2. Thermodynamic properties Eqs. (3)–(6) are used to plot the temperature dependence of the internal energy (E), free energy (F), entropy (S), and heat capacity (C) at constant volume for each compound by using the data obtained from VASP and Phonon software. These predicted high temperature behaviors could give some information when XB2

Total Ce B

8

Free Energy [meV/u.c.x100]

Int. Energy [meV/u.c.x100]

9

compound annealed at different temperatures to improve their performance. The temperature variation of thermodynamical functions exhibits, almost, similar trend for the considered XB2 compounds. Therefore, the temperature dependence of the thermodynamical properties are only given for CeB2 compound (see Fig. 3). Unfortunately, there are no experimental and theoretical data available to check our results. The predicted internal energies for CeB2 in the AlB2-type structure as a function of temperature displayed in Fig. 3a suggest that, above 300 K, the internal energies increase almost linearly with temperature and tend to the kBT behavior for all compounds. The total and partial internal energy graphs exhibit similar trend, and the contribution to internal energy from B atoms is more dominant than from X atoms. Fig. 3b shows the free energy versus temperature plot with similar characteristics and free energy decreases gradually with increasing temperature. This behavior is due to the fact that the both internal energy and entropy increase with temperature, and this leads to the decreasing in free energy [48]. Also, the inspection of the free energy plots suggests that these compounds are thermodynamically stable in the studied temperature range. The variations of entropy with the temperature for studied structure of CeB2 are given in Fig. 3c for the same temperature range. Obviously, at 0 K, their entropies are zero and the entropy change increase rapidly as temperature increasing at low temperature, while the variation of entropy is small above about 300 K. The entropy of a crystal is always caused by electronic excitation and lattice vibration, and the entropies will increase as the temperature increases [49]. The calculated values of entropy at 300 K and 1000 K are listed in Table 2. The difference between these values could be the responsible for increasing of the maximum value of the phonon

7 6 5 4 3 2

(a)

1

Heat Capacity [kB/u.c.]

Entropy [kB/u.c.]

-4 -6 -8

Total Ce B

10

Total Ce B

16

-2

-10

0

18

0

14 12 10 8 6 4 2

(b)

Total Ce B

8

6

4

2

(c)

(d)

0

0 0

200

400

600

800

Temperature [K]

1000

1200

0

200

400

600

800

1000

Temperature [K]

Fig. 3. Temperature dependence of internal energy, free energy, entropy, and heat capacity of CeB2 in the AlB2-type structure.

1200

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H. Ozisik et al. / Computational Materials Science 68 (2013) 307–313

235

Free Energy [meV/u.c.]

230 225 220 215 210 205 200 195 190

2

2

Pm B Sm 2 B 2 Eu B 2 G dB 2 Tb B 2 D yB 2 H oB 2 Er B 2 Tm B Lu 2 B

2

dB N

C

eB 2 Pr B

185

Fig. 4. Trends of the calculated free energy at 0 K.

frequencies from CeB2 to LuB2 (see Figs. 2 and 3). These differences may stem from the higher contribution of the low phonon frequency [50]. The low-frequency modes which have longer wavelengths are associated with larger volumes in the configurational space. Therefore, they cause higher values for specific heat and entropy [50]. From Fig. 3c, one can see that at low temperature (about <800 K) Ce atoms has higher entropy than that of B atoms. However, an opposite situation occurs when the temperature is increased (about >800 K), and the contribution to entropy from B atoms is more dominant. In generally, it is worth stating that the entropy decreases when the size of the cation mass increases (Ce ? Lu). The contributions to the total heat capacity from the lattice vibrations of CeB2 illustrated in Fig. 3d. This figure suggests that while the temperature is about T < 400 K, Cv increases very rapidly with the temperature; when the temperature is about T > 400 K, Cv increases slowly with the temperature and it almost approaches to a constant called Dulong–Petit limit. Moreover, these compounds obey the Debye T3 law at low temperatures. The calculated values of heat capacity at 300 K and 1000 K are listed in Table 2. From Fig. 3d, one can see that at low temperature (about <200 K) Ce atoms has higher heat capacity than that of B atoms. However, an opposite situation occurs when the temperature is increased (about >200 K), and the contribution to entropy from B atoms is more dominant. The calculated values of free energy at 0 K are given in Table 2 and demonstrated in Fig. 4. It can be seen from Figs. 2b and 4, the calculated free energies and maximum value of the phonon frequencies are exhibits, almost, similar trend. It is worth mentioning that the anion atoms are the same but the cation atom’s size is different in these compounds. This difference could be the responsible for increasing of the maximum value of the phonon frequencies and free energies from CeB2 to LuB2. In Figs. 2b and 4, the trend of free energy and maximum phonon frequency from Ce to Gd are close to each other, but both of them increases more rapidly from Gd to Lu in accordance with their atomic number. The free energies are calculated in terms of phonon frequencies and total density of states (Eq. (4)). Therefore, the temperature dependent free and internal energies are higher for LuB2 than other compounds since LuB2 has much higher average phonon frequencies and lower entropy.

4. Conclusions In this paper, we have investigated the lattice dynamical and thermodynamical properties of XB2 compounds using the density

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