The structural, electronic, elastic, vibrational, and thermodynamic properties of HoX (X=Sb, Bi)

Physica B 405 (2010) 3977–3985

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Physica B journal homepage: www.elsevier.com/locate/physb

The structural, electronic, elastic, vibrational, and thermodynamic properties of HoX (X ¼ Sb, Bi) ¨ ztekin C Cansu C - oban a,n, Kemal C - olako˘glu b, Yasemin O - iftc- i b a b

Balıkesir University, Department of Physics, 10145 C - a˘gıs- , Balıkesir, Turkey Gazi University, Department of Physics, Teknikokullar, 06500 Ankara, Turkey

a r t i c l e in fo

abstract

Article history: Received 7 July 2009 Received in revised form 23 May 2010 Accepted 23 June 2010

We have predicted the structural, electronic, elastic, vibrational, and thermodynamic characteristics of HoX (X ¼ Sb, Bi) compounds in NaCl type (B1) structure through the method of density functional theory within the generalized gradient approximation (GGA) using Vienna Ab initio Simulation Package (VASP). Specifically, the lattice constant, bulk modulus, pressure derivative of bulk modulus, cohesive energy, second-order elastic constants, Young’s modulus, isotropic shear modulus, Zener anisotropy factor, Poisson’s ratio, electronic band structures, and related total density of states (DOS) have been calculated and compared with the available experimental data. The spin–polarization (SP) and spin– orbit coupling (SOC) have been taken into account for the electronic structural calculations. The phonon frequencies and one-phonon DOS have also been calculated and presented. In order to gain further information, the pressure and temperature dependent behaviour of the volume, bulk modulus, thermal ¨ expansion coefficient, heat capacity, entropy, Debye temperature, and Gruneisen parameter have been evaluated over a pressure range 0–25 GPa and a wide temperature range 0–1600 K. & 2010 Elsevier B.V. All rights reserved.

Keywords: HoSb and HoBi Elastic properties Lattice dynamics Thermodynamic properties

1. Introduction Rare-earth elements have recently attracted much interest due to the mechanisms for magnetic ordering, the pressure/strain and impurity effects, and unusual structural, electronic, and highpressure properties. While the rare-earth ion is normally in the trivalent state, the rare-earth pnictides with N, P, As, Sb, and Bi atoms crystallize in the NaCl type (B1) structure [1]. Rare-earth pnictides are typically low carrier and strongly correlated systems [2]. The presence of 4f electrons in these compounds is mainly responsible for the various magnetic and electrical properties [3]. HoX (Ho:0, 0, 0; X¼ Sb, Bi:1/2, 1/2, 1/2; space group Fm3¯m (2 2 5)) compounds with B1 structure are members of the rareearth pnictides. A few experimental studies are available for HoSb and HoBi in the literature. Shirotani et al. [4] have systematically investigated pressure-induced phase transitions in lanthanide monoantimonides with a B1 structure by measuring the powder X-ray diffraction pattern using synchrotron radiation. Yoshihara et al. [5] have measured the lattice parameters of Ho-rich and Bi-rich HoBi samples that had been annealed at 700 and 900 1C. They have reported a metal-rich phase that they designated Ho5 xBi3.

The melting point of HoBi was estimated by Kovenskaya et al. [6] using an empirical method and they have calculated the Debye temperature (Y) from these results. Abdusalyamova et al. [7] have also investigated the Debye temperature, Y, using the thermal expansion coefficient data. Parodi et al. [8] have made some calorimetric measurements on the HoBi systems. Andersen et al. [9] have studied the temperature variation of magnetic structure using neutron diffraction technique in the antiferromagnet HoSb. Jensen et a1. [10] have made an extensive study of the influence of such anisotropic interactions on the magnetic properties of HoSb. Some elastic and thermodynamic measurements have been conducted on some rare-earth antimonides including the HoSb by Mullen et al. [11]. To our knowledge, there has been no theoretical investigation on the electronic, vibrational, and thermoelastic properties of these compounds up to now. In the present study, we have primarily focused on the structural, electronic, vibrational, and thermo-elastic properties of these compounds to add some information to the existing data. The spin–polarization (SP) and spin–orbit coupling (SOC) interactions have been taken into account in the lattice constant, electronic band structure, and total density of states calculations.

2. Method of calculation n

Corresponding author. Tel.: +90 266 6121278/1206; fax: + 90 266 6121215. E-mail addresses: [email protected] (C. C - oban), [email protected] ¨ ztekin C (K. C - olako˘glu), [email protected] (Y. O - iftc- i). 0921-4526/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2010.06.042

All calculations conducted in this paper were done within the Vienna Ab initio Simulation Package (VASP) [12–16]. It performs

C. C - oban et al. / Physica B 405 (2010) 3977–3985

an iterative solution of the generalized Kohn–Sham equations of density-functional theory, based on the minimization of the norm of the residual vector to each eigenstate and an efficient charge density mixing [17]. The gradient-corrected functionals in the form of the generalized-gradient approximation (GGA) developed by Perdew and Wang [18,19] have been chosen. The projector¨ augmented wave (PAW) method developed by Blochl [20] were implemented within VASP is used to describe the interactions between ions and electrons. All the results presented below have been obtained by using a plane wave basis with an energy cutoff of 500 eV. The k-points of 12  12  12 have been generated by using Monkhorst and Pack scheme [21] for the Brillouin zone sampling. The present GGA phonon frequencies of HoSb and HoBi compounds have been calculated by the PHON program [22] using the forces based on the VASP package. The PHON code calculates phonon frequencies and one-phonon density of states (DOS) by generating all elements of the force constant matrix. It uses the ‘‘Small Displacement Method’’ described in Ref. [23] which is a similar procedure as described in Ref. [24]. Specifically, our phonon calculations have been calculated in high symmetry directions using a 2  2  2 cubic supercell of 48 atoms. The quasi-harmonic Debye model has been applied to the calculation of thermodynamic properties of HoX (X¼ Sb, Bi) compounds. The non-equilibrium Gibbs function G*(V; P, T) within this model is defined as [25]

where n is the number of atoms per formula unit and D(Y/T) represents the Debye integral. Y is expressed as [28] rffiffiffiffiffi i1=3 _h Bs Y ¼ 6p2 V 1=2 n f ðsÞ ð3Þ k M where M is the molecular mass per unit cell, s is the Poisson ratio and BS is the adiabatic bulk modulus [31]. f(s) is taken from Refs. [28,29]. Therefore, the non-equilibrium Gibbs function G*(V; P, T) as a function of (V; P, T) can be minimized with respect to V as in following equation:   @G*ðV; P,TÞ ¼0 ð4Þ @V P,T By solving Eq. 4, one can obtain the thermal equation of state (EOS) V(P, T). The entropy S, heat capacity at constant volume CV, ¨ isothermal bulk modulus BT, Gruneisen parameter g, thermal expansion coefficient a, and heat capacity at constant pressure CP of the system are given as in Eqs. (5–10) [31] h i S ¼ nk 4DðY=TÞ3lnð1eY=T Þ ð5Þ

@2 G*ðV; P,TÞ BT ðP,TÞ ¼ V @V 2

g¼

d lnYðVÞ d ln V

ð9Þ

CP ¼ CV ð1 þ agTÞ

ð10Þ

3. Results and discussion 3.1. Structural and electronic properties We have computed the equilibrium lattice constant with and without SP. The bulk modulus and its pressure derivative have been calculated minimizing the total energy for B1 structure of the HoX (X¼Sb, Bi) at different volumes by means of Murnaghan’s equation of state [32]. Optimized lattice constants without SP for these compounds are listed in Table 1 along with their experimental values taken from Refs. [11] and [33] for HoSb, [5] and [33] for HoBi. They are found to be 6.18 and 6.31 A˚ for HoSb and HoBi, respectively. For the SP case, our values (6.16 A˚ for HoSb and 6.296 A˚ for HoBi) are given in the same table which are closer to the experimental data given in Table 1 than lattice constants Table 1 Calculated lattice constant (a), bulk modulus (B), pressure derivative of bulk modulus (B’), and cohesive energy (Ecoh ) for B1 structure of HoX (X ¼ Sb, Bi).

ð1Þ

where E(V) is the total energy for per unit cell of HoX (X¼Sb, Bi), PV is the constant hydrostatic pressure condition, Y(V) is the Debye temperature, and Avib is the vibrational Helmholtz free energy which is given below [26–30]   9Y þ 3lnð1eY=T ÞDðY=TÞ ð2Þ Avib ðY; TÞ ¼ nkT 8T

  3Y=T CV ¼ 3nk 4DðY=TÞ y=T e 1

gCV BT V

ð6Þ

! ð7Þ P,T

ð8Þ

Material

Reference

˚ a (A)

B (GPa)

B’

Ecoh (eV/atom)

HoSb

Present (GGA) Present (GGA+ SP) Experimenta Experimentb Present (GGA) Present (GGA+ SP) Experimentc Experimentb

6.18 6.16 6.13 6.12 6.31 6.296 6.228 6.215

60.22

3.86

5.44

54.14

3.99

4.74

HoBi

a b c

Energy (eV)

G*ðV; P,TÞ ¼ EðVÞ þ PV þ Avib ½YðVÞ; T 

a¼

Energy (eV)

3978

Ref. [11]. Ref. [33]. Ref. [5].

4 0 -4 -8 -12 -16 -20 -24 HoSb -28 L

4 0 -4 -8 -12 -16 -20 -24 HoBi -28 L

EF

Γ

X

W

5 10 15 DOS (States/eV)

EF

Γ

X

W

5 10 15 DOS (States/eV)

Fig. 1. Calculated electronic band structures and total density of states without SP for B1 phase of (a) HoSb and (b) HoBi.

C. C - oban et al. / Physica B 405 (2010) 3977–3985

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materials exhibit nearly a semi-metallic character. The main difference between bands with SOC and without SOC is the accession in the values of the band energies. The total DOS corresponding to the present band structures have also been calculated taking into account the SP and SOC (see the same Figs.). The absence of the energy-gap in DOS profiles again supports the metallic nature of the considered compounds. But, surprisingly, the expected splittings and shifts in the bands and DOS profiles for the spin-polarized calculations are not observed. The band states almost coincide in undistinguishable manner for the spinup/spin-down cases (see Fig. 2a,c).

without SP. The computed bulk modulus and its pressure derivative of HoX (X¼Sb, Bi) are also listed in this table. Unfortunately, the previous works related to bulk modulus and its pressure derivative are not available to compare with the present results. The cohesive energy (Ecoh) of a given phase is defined as the difference in the total energy of the constituent atoms at infinite separation and the total energy of that particular phase h i ð11Þ EAB coh ¼ EA atom þEB atom EAB total where EAB total is the total energy of the HoX (X¼Sb, Bi) at equilibrium lattice constant and EAatom and EBatom refer to the atomic energies of the pure constituents. The computed cohesive energies, presented in Table 1, are estimated to be 5.44 and 4.74 eV/ atom for B1 structure of HoSb and HoBi compounds, respectively. Present results decrease on going from Sb to Bi and they are close to those found for SmSb and SmBi as in our recent work [34]. Although it is not our main purpose to do the detailed band structure calculations, the electronic band structures of HoX (X¼Sb, Bi) along with the high symmetry directions have been calculated for B1 phase by using the calculated equilibrium lattice constant in the absence and in the presence of SP and SOC (see Figs. 1–3). The band gap is not observed for B1 phase of these compounds for both with and without SP cases, and these

3.2. Elastic properties The elastic constants are important properties of solids which provide a link between the mechanical and dynamic behaviours of crystals. In particular, they provide information on the elasticity, stability, and stiffness of crystals and give important information concerning the nature of the forces operating in solids. Their ab initio calculation requires precise methods. Because the forces and the elastic constants are functions of the first- and second-order derivatives of the potential, their calculation will provide a further check on the accuracy of the calculation

EF

8

4

6

EF

0

HoSb (SP)

4 DOS (States/eV)

Energy (eV)

-4 -8 -12 -16 spin_down

HoSb (SP)

-2 spin_up spin_down

-6

-24

-8 Γ

L

X

-20

W

-15

-10

-5 0 Energy (eV)

10

HoBi (SP)

6

EF

0

5

EF

8

4

4 DOS (States/eV)

-4 Energy (eV)

0

-4

spin_up

-20

2

-8 -12 -16

0 -2 -4

spin_up

-20

2

spin_down

spin_up spin_down

-6

HoBi (SP)

-24

-8

L

Γ

X

W

-20

-15

-10

-5 0 Energy (eV)

Fig. 2. Calculated spin-polarized electronic band structures and total density of states for B1 phase of (a), (b) HoSb and (c), (d) HoBi.

5

10

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15 10 5 0 -5 -10 -15 -20

350

HoSb (SOC)

Γ

Energy (eV)

L

16 12 8 4 0 -4 -8 -12 -16

X

W

2

4 6 8 10 12 DOS (States/eV)

250 200 150 C11

100

EF

C12

HoSb

C44

0 0

5

10 15 Pressure (GPa)

HoBi (SOC)

Γ

X

W

2

4 6 8 10 DOS (States/eV)

Fig. 3. Calculated electronic band structures and total density of states with SOC for B1 phase of (a) HoSb and (b) HoBi.

Table 2 Elastic constants and Cauchy pressure (C12–C44) (in GPa) for HoX (X ¼Sb, Bi) in B1 structure. Material

Reference

C11

C12

C11–C12

C44

C12–C44

HoSb

Present Experimenta Present

150.23

17.48

132.75 138.00 116.39

25.25 27.60 21.73

 7.77

a

300

50

L

HoBi

Elastic constant (GPa)

EF

133.05

16.66

25

300 250 200 150 C11

100

C12

HoBi

50

 5.07

C44

0

Ref. [11].

of forces in solids. A study of the elastic properties for materials is also essential to understand the chemical bonds and the cohesion of material. To realize these purposes, the second-order elastic constants have been computed using the ‘‘stress–strain’’ relations [35] in a manner similar to our recent work [36] and the results are listed in Table 2 along with the experimental values of Ref. [11]. The traditional mechanical stability conditions in cubic crystals on the elastic constants are known as C11–C12 40, C11 40, C44 4 0, C11 +2C12 40, and C12 oB oC11. Our results for elastic constants satisfy these stability conditions and they are consistent with the experimental values of Mullen et. al. [11] for HoSb. The pressure dependency of the second-order elastic constants, up to 25 Gpa, have also been computed (see Fig.4) in order to see the effect of pressure on the elastic constants. The C11, C12, and C44 are positive in 0–25 GPa pressure range. While C11 increases monotonically with increasing pressure, C12 and C44 decrease under the same pressure range. Here, an unexpected violation is observed on the Cauchy relation C12–C44 ¼2P (P: Pressure), i.e., our calculations give negative C12–C44 values (see Table 2). C44 4C12 inequality is observed both in the present and experimental work [11]. According to Pettifor [37], the angular character of atomic bonding in metals and compounds can be explained in terms of the negative Cauchy pressure, i.e., the Cauchy pressure may become negative for the directional bonding in some ductile (such as Ni and Al)/brittle (such as Si) materials. Also, it is thought that the negative Cauchy discrepancy is a consequence of the hybridization of the unstable ‘f’ band, and hybridization may be responsible for the decrease in Ho–Ho distance and thereby a small value of the C12 [38].

20

350 Elastic constant (GPa)

Energy (eV)

3980

0

5

10

15

20

25

Pressure (GPa) Fig. 4. The pressure dependence of the elastic constants of (a) HoSb and (b) HoBi.

Table 3 The calculated Zener anisotropy factor (A), Poisson’s ratio (n), Young’s modulus (E), and isotropic shear modulus (G) for HoX (X ¼ Sb, Bi) in B1 structure. Material

Reference

A

n

E (GPa)

G (GPa)

HoSb HoBi

Present Present

0.38 0.37

0.24 0.25

93.44 81.57

37.63 32.66

For the investigation of their hardness, the elastic properties e.g. the Zener anisotropy factor (A), Poisson’s ratio (n), and Young’s modulus (E) are measured for polycrystalline materials. The Zener anisotropy factor, A, is calculated using the relation given as [39] A¼

2C44 C11 C12

ð12Þ

Poisson’s ratio, n, and Young’s modulus, E, are calculated in terms of the computed data using the following relations [40]:

n¼

" # 1 B 23 G   2 B þ 13 G

ð13Þ

C. C - oban et al. / Physica B 405 (2010) 3977–3985

and

Frequency (cm-1)

40 35

E¼

30 25

9GB G þ 3B

ð14Þ

where G ¼(GV +GR)/2 is the isotropic shear modulus, GV is Voigt’s shear modulus corresponding to the upper bound of G values, and GR is Reuss’s shear modulus corresponding to the lower bound of G values; they can be written as

20 15 10 5

HoSb

0

GV ¼ ðC11 C12 þ3C44 Þ=5 Γ

L

X

W

25 20 15 10 5 HoBi

0

Γ

L

X

W

and

5=GR ¼ 4=ðC11 C12 Þ þ3=C44

DOS

30 Frequency (cm-1)

3981

DOS

Fig. 5. Calculated phonon dispersion curves and one-phonon density of states for (a) HoSb and (b) HoBi in B1 structure.

The calculated Zener anisotropy factor, A, Poisson’s ratio, n, Young’s modulus, E, and isotropic shear modulus, G, for HoX (X¼Sb, Bi) are presented in Table 3. The Zener anisotropy factor, A, is a measure of the degree of elastic anisotropy in solids. The A takes the value of 1 for a completely isotropic material. If the value of A is smaller or greater than 1 it shows the degree of elastic anisotropy. The calculated Zener anisotropy factors for HoX (X¼ Sb, Bi) are smaller than 1 which indicates that these compounds have an elastically anisotropic character. The Poisson’s ratio, n, is very important property for industrial applications. It is small (n ¼0.1) for covalent materials, and it has a typical value of n ¼0.25 for ionic materials [41]. Calculated n

460 T=0 K T=400 K T=800 K T=1200 K T=1600 K

Volume (Bohr3)

400

P=0 GPa P=15 GPa

440

P=5 GPa P=20 GPa

P=10 GPa P=25 GPa

420 Volume (Bohr3)

420

380 360

400 380 360 340

340 HoSb

320

320 HoSb 300

0

5

10

15

20

25

0

200

400

Pressure (GPa) 480

460 T=0 K T=400 K T=800 K T=1200 K T=1600 K

420

P=0 GPa P=15 GPa

460

P=5 GPa P=20 GPa

P=10 GPa P=25 GPa

440 Volume (Bohr3)

440 Volume (Bohr3)

600 800 1000 1200 1400 1600 Temperature (K)

400 380 360

420 400 380 360

HoBi

340

340 HoBi

320 0

5

10

15

Pressure (GPa)

20

25

0

400

800 Temperature (K)

Fig. 6. The pressure and temperature dependence of the volume of (a), (b) HoSb and (c), (d) HoBi.

1200

1600

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3982

values are equal to 0.24 and 0.25 for HoSb and HoBi, respectively. It shows that, the ionic contributions to the atomic bonding are dominant for these compounds. The Young’s modulus, E, which is the ratio of the tensile stress to the corresponding tensile strain, is required to provide information about the measure of the stiffness of the solids. The material is stiffer for the larger value of Young’s modulus. The present values of Young’s moduli decrease from Sb to Bi, which indicates that HoSb is stiffer than HoBi. We can also estimate the brittle/ductile behaviour of these compounds using the calculated bulk modulus and isotropic shear modulus. According to the criterion, proposed by Pugh [42], a material is brittle if the B/G ratio is less than 1.75, otherwise it behaves in ductile manner. The present values of B/G are found to be 1.600 for HoSb and 1.658 for HoBi. In our case, these parameters are less than 1.75, i.e. these materials will behave in a brittle manner.

properties such as heat capacities, thermal expansion coefficients, vibrational entropy, etc. For these purposes, the phonon dispersion curves and one-phonon DOS without LO/TO splitting of HoSb and HoBi have been calculated and displayed in Fig. 5. The acoustic and optical branches are well separated in crystals when the mass difference between the two atoms in the unit cell is large. Here, a small gap between the acoustic and optical branches is only observed in the G X direction for about qmax/2 wave vectors. The shape of the dispersion curves slightly changes depending on the mass difference between pnictide ions. The maximum values of the phonon frequencies for acoustic branches decrease on going from Sb to Bi. Additionally, the soft modes (negative frequencies) are not observed for any wave vector for the considered phase at ambient conditions, which strongly supports the dynamic stability of B1 structure. There are no experimental or other theoretical data for the comparison with our calculated results.

3.3. Vibrational properties

3.4. Thermodynamic properties

It is well known that the energy dispersion of phonons in solids provides rich information about the dynamic properties. In particular, it is an essential input in calculation of thermodynamic

The pressure and temperature dependence of the volume and bulk modulus of HoX (X¼Sb, Bi) have been investigated using the quasi-harmonic Debye model up to 1600 K. One can obviously see

P=0 GPa P=15 GPa

160

P=5 GPa P=20 GPa

P=10 GPa P=25 GPa

Bulk modulus (GPa)

140 Bulk modulus (GPa)

HoSb

140

120 100 80

120 100 T=0 K T=400 K T=800 K T=1200 K T=1600 K

80 60

60 HoSb 0

200

400

600

800

1000 1200 1400 1600

0

5

Temperature (K)

P=0 GPa P=15 GPa

160

P=5 GPa P=20 GPa

15

20

25

Pressure (GPa)

P=10 GPa P=25 GPa

140

140

HoBi

120 Bulk modulus (GPa)

Bulk modulus (GPa)

10

120 100 80

100 T=0 K T=400 K T=800 K T=1200 K T=1600 K

80 60

60 HoBi

40 0

40 400

800 Temperature (K)

1200

1600

0

5

10

15

Pressure (GPa)

Fig. 7. The temperature and pressure dependence of bulk modulus of (a), (b) HoSb and (c), (d) HoBi.

20

25

C. C - oban et al. / Physica B 405 (2010) 3977–3985

from Fig. 6 that, for these temperatures, the volume decreases dramatically as P increases. At 0 GPa, the volume increases rapidly with the temperature. Under higher pressure, the volume varies gradually as the temperature rises. These changes can be attributed to the stronger atomic interactions in the interlayer. Fig. 7(a) and (c) show the variations of bulk modulus B with the temperature (0–1600 K) for HoSb and HoBi, respectively. The relationship between bulk modulus B and pressure P at different temperatures, T¼0, 400, 800, 1200, and 1600 K, are also presented in Fig. 7(b) and (d) for HoSb and HoBi, respectively. In Fig. 7, the bulk modulus B decreases gradually as T increases at a given pressure which indicates that the cell volume undergoes gradual changes and increases rapidly as P increases at a given temperature. We can say that, the effect of increasing pressure is the same as that of the decreasing temperature on HoX (X¼Sb, Bi). Using the quasi-harmonic Debye model described above, the values of heat capacity at constant volume, CV, and at constant pressure, CP, can be determined. Calculated relationships between heat capacities (CV and CP) and temperatures at different pressures are presented in Fig. 8. As we can see from Fig. 8, when To400 K CV increases rapidly with temperature at a given pressure and it decreases with pressure at a given temperature. Due to the use of anharmonic approximation of the Debye model,

3983

the heat capacities CV and CP exhibit strong dependency on temperature T and pressure P for To400 K. For temperatures higher than 400 K, the influence of anharmonicity on CV is less than on CP. At high temperatures, CP increases linearly whereas the CV approaches the Dulong–Petit limit (CV(T)  3 R) as shown in Fig. 8. This behaviour shows that the atomic interactions are dominant at low temperatures in these compounds. The variations of entropy with the temperature at different pressures for HoX (X¼ Sb, Bi) compounds are presented in Fig. 9. Obviously, entropy increases with increasing temperature but it decreases with the pressure. The variations of thermal expansion coefficient, a, with different temperatures at different pressures have been plotted in Fig. 10. This figure shows that at low temperatures (To250 K) a increases rapidly with temperature. Relatively at high temperatures (T4250 K), a linear increase occurs for HoSb at all pressures. But, at P¼0 GPa, when T¼1250 K it starts to exhibit decreasing trend with increasing temperature for HoBi. It may be noted that as the pressure increases, the increase of a with temperature becomes smaller. The Debye temperature, Y, is closely related to the elastic constants, specific heat, and melting temperature. This important physical parameter has been calculated in quasi-harmonic approximation and the results are presented in Table 4 along

60 168

Dulong-Petit Limit

147 Entropy (Jmol K )

40

-1 -1

-1 -1

Heat capacity (Jmol K )

50

30 20 10

CV 0 GPa

CV 5 GPa

CV 10 GPa

CV 15 GPa

CV 20 GPa

CV 25 GPa

CP 0 GPa

CP 5 GPa

CP 10 GPa

CP 15 GPa

CP 20 GPa

CP 25 GPa

0 400

800 Temperature (K)

1200

1600

63 P=0 GPa P=15 GPa

42

P=5 GPa P=20 GPa

P=10 GPa P=25 GPa

HoSb

0

400

800 Temperature (K)

1200

1600

180

Dulong-Petit Limit

160

50

140 -1 -1

Entropy (Jmol K )

-1 -1

84

0

60

Heat capacity (Jmol K )

105

21

HoSb 0

126

40 30 20 10

CV 0 GPa

CV 5 GPa

CV 10 GPa

CV 15 GPa

CV 20 GPa

CV 25 GPa

CP 0 GPa

CP 5 GPa

CP 10 GPa

CP 15 GPa

CP 20 GPa

CP 25 GPa

120 100 80 60

P=0 GPa P=15 GPa

40

P=5 GPa P=20 GPa

HoBi

20

HoBi 0

P=10 GPa P=25 GPa

0

0

400

800 Temperature (K)

1200

1600

Fig. 8. The variations of heat capacities CV and CP with the temperature at different pressures for (a) HoSb and (b) HoBi.

0

400

800 Temperature (K)

1200

1600

Fig. 9. Temperature dependence of entropy at different pressures for (a) HoSb and (b) HoBi.

C. C - oban et al. / Physica B 405 (2010) 3977–3985

3984

Table 4 ¨ The variations of calculated Debye temperature Y (K) and the Gruneisen parameter g with different temperatures and pressures for HoX (X ¼ Sb, Bi) compounds.

Thermal expansion coefficient (10-5K-1)

7 P=0 GPa P=15 GPa

6

P=5 GPa P=20 GPa

P=10 GPa P=25 GPa

Reference

5 4

Material

1

P (GPa)

Y (K)

g

Y (K)

g

Present

0

Present

400

Present

800

Present

1200

Present

1600

0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25

231.82 263.57 289.77 312.03 331.47 349.02 225.32 258.76 285.94 308.86 328.76 346.54 217.29 252.60 280.73 304.49 325.30 343.10 209.14 245.60 275.49 300.21 321.18 339.59 200.65 238.51 269.54 295.23 316.99 336.01

1.882 1.749 1.635 1.542 1.466 1.402 1.905 1.770 1.651 1.555 1.477 1.411 1.930 1.797 1.674 1.573 1.490 1.423 1.950 1.827 1.697 1.591 1.506 1.436 1.961 1.856 1.723 1.612 1.522 1.449

189.36 225.36 249.20 269.39 286.98 302.87 183.30 220.79 245.54 266.37 284.40 300.49 176.04 215.08 240.76 262.36 281.25 297.33 168.82 208.76 235.93 258.45 277.48 294.12 162.03 202.35 230.48 253.88 273.65 290.86 170.00 143.00

1.976 1.833 1.717 1.622 1.543 1.476 1.988 1.854 1.735 1.636 1.554 1.486 1.992 1.880 1.758 1.655 1.568 1.499 1.981 1.908 1.782 1.673 1.585 1.512 1.951 1.934 1.808 1.695 1.602 1.526

HoSb

0 0

400

800 Temperature (K)

1200

1600

Thermal expansion coefficient (10-5K-1)

7 P=0 GPa P=15 GPa

6

P=5 GPa P=20 GPa

P=10 GPa P=25 GPa

5 4 3 Experimenta Experimentb Experimentc

2 1

HoBi

a b

0 0

400

800 Temperature (K)

1200

1600

Fig. 10. The thermal expansion coefficient versus temperature curves at different pressures for (a) HoSb and (b) HoBi.

with the experimental ones. The values of Y decrease with the temperature, but increase as the pressure increases. When the temperature is constant, the change in the Y increases almost linearly with the pressure (from P410 GPa). The Y of HoBi have been calculated to be 170 K from melting point data by Kovenskaya et al. [6] and 143 K from thermal expansion data by Abdusalyamova [7]. From Table 4, at 0 GPa our present values for HoBi are closer to the data taken from Ref. [6]. At 1200 K, this agreement is better than the others. The present value of Y for HoSb is consistent with the value of Ref. [11]. Both compounds exhibit a similar trend under pressure and temperature variations. The calculated properties at different temperatures are very sensitive to the vibrational modes. In the quasi-harmonic Debye ¨ model, the Gruneisen parameter, g, is a key quantity. It could describe the alteration in vibration of a crystal lattice based on the increase or decrease in volume as a result of temperature change [43]. The only explicit dependence of this parameter is on V [31]. However, it has an implicit dependence on T and P [31]. In Table 4, ¨ the Gruneisen parameter is also given for the various temperatures at P¼ 0, 5, 10, 15, 20, and 25 GPa. As can be seen, the calculated g slightly changes with temperature and in practice, one can think that it remains constant. This quantity, g, decreases more rapidly with pressure for both compounds which implies

HoBi

T (K)

3 2

HoSb

c

247.00

Ref. [6]. Ref. [7]. Ref. [11].

that the vibrational modes in HoX (X¼Sb, Bi) are similar to one another. At constant temperature, the g values for HoSb are smaller than that for HoBi under the same pressure. Hence, it is worth noting that they are sensitive to the Sb and Bi ions in these compounds.

4. Summary and conclusion In summary, the first-principles calculations have been performed in order to explore the structural, electronic, elastic, lattice dynamic, and thermodynamic properties of HoX (X¼ Sb, Bi) compounds in B1 structure. The obtained lattice constants are in good agreement with the available experimental data. The calculated elastic constants satisfy the traditional mechanical stability conditions and agree with the available experimental values, but violate the Cauchy relation. The phonon dispersion curves without soft mode strongly support the dynamic stability of B1 phase. The results on the temperature and/or pressure dependence of the various thermodynamic quantities such as bulk modulus, specific heats, thermal expansion coefficient, ¨ Debye temperature, and Gruneisen parameter providing valuable information on the intrinsic character of solids are also presented. We conclude that these theoretical results, in particular, on the mechanical, vibrational and thermo-elastic behaviour of HoSb and

C. C - oban et al. / Physica B 405 (2010) 3977–3985

HoBi are reliable reference to further experimental and theoretical investigations.

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