Ab initio calculations of the thermodynamicsand phase diagram of ScSb

Min Teng • Hongzhi Fu • Jianwu Feng •

WenFang Liu • Gao Tao

Received: 9 January 2012 / Accepted: 22 May 2012 / Published online: 1 June 2012

� Springer Science+Business Media, LLC 2012

Abstract The hydrostatic pressure-induced transition

phase of ScSb from the sixfold-coordinated NaCl-type (B1)

to the eightfold-coordinated CsCl-type (B2) is investigated

within the Perdew–Burke–Ernzerhof (PBE) form of gen-

eralized gradient approximation. It is found that the tran-

sition pressure from the B1 to B2 phase is at 42.8 GPa

which agrees with experiment. Through the quasi-har-

monic Debye model, the dependences of the relative vol-

ume V on the pressure P, the thermal expansion, the

Gruneisen parameter ratio, (c - c0)/c0, the Debye tem-

perature H, and heat capacity CV on the pressure P and

temperature T are estimated. The quasi-harmonic Debye

model for the first time is used to predict phase diagram of

B1 ? B2.

Introduction

As one of the group-III–V transition metals, the scandium

compounds have recently attracted particular interest

because of various anomalous physical properties, in

respect of structural, magnetic, and phonon properties [1].

By use of synchrotron radiation, Hayashi et al. [2] have

studied powder X-ray diffraction of ScSb and YSb with the

NaCl-type structure up to 45 GPa at room temperature and

found the B1–B2 transition at high pressures. ScSb and

YSb compounds crystallize in B1 structure at ambient

conditions. Under pressure, ScSb have been found to begin

a first-order phase transition from sixfold-coordinated

NaCl-type (B1) to the eightfold-coordinated CsCl-type

(B2) at around 28 GPa. The low- and high-pressure

structures coexist between 28 and 43 GPa. For YSb, the

low- and high-pressure structures coexist between 26

and 36 GPa [2]. Previous theoretical study using a

Trouiller–Martins non-local norm-conserving pseudopo-

tentials predicted the transition phase pressures to be 16

and 22.5 GPa for ScSb and YSb, respectively [3].

Recently, Chen et al. [1], theoretically studied the high

pressure-induced phase transition of YSb and ScSb com-

pounds using DFT–GGA method. It was found that the

phase transition from the B1 to B2 structures began to

occur at around 29 and 40 GPa for YSb and ScSb com-

pounds, respectively. Bouhemadou and Khenata [4] and

Maachou et al. [5] investigated the structural, elastic, and

high pressure properties in ScSb, using the full-potential

augmented plane wave plus local orbital’s method

(FP-APW ? lo), and they show the phase transitions of

ScSb from B1 to B2 at 39.78 GPa. In view to the inter-

action potential with long-range Coulomb, van der Waals

interaction and the short-range repulsive interaction up to

second-neighbor ions within the Hafemeister and Flygare

M. Teng

Department of Information Technology, Henan Judicial Police

Vocational College, Zhengzhou 450011, People’s Republic

of China

H. Fu (&) � J. Feng

College of Physics and Electronic Information, Luoyang Normal

College, Luoyang 471022, People’s Republic of China

e-mail: fhzscdx@163.com

H. Fu

National Laboratory of Superhard Materials, Jilin University,

Changchun 130012, People’s Republic of China

W. Liu

College of Chemistry and Chemical Engineering, Luoyang

Normal College, Luoyang 471022, People’s Republic of China

G. Tao

Institute of Atomic and Molecular Physics, Sichuan University,

Chengdu 610065, People’s Republic of China

123

J Mater Sci (2012) 47:6673–6678

DOI 10.1007/s10853-012-6605-x

approach, Varshney et al. [6] analyze the structural as well

as elastic properties in yttrium and scandium antimonides.

They concluded that the observed structural transformation

shall be responsible for the transfer of electrons from Sb s

and p like states to the Y(Sc)-d like states continuously

under pressure.

Due to the absence of theoretical calculations and

experimental measurements on the thermodynamic prop-

erties of ScSb and in order to study its equation of states

(EOSs), we have performed first-principles study of its

structural and thermodynamic properties under high pres-

sure. Our main objectives in this paper are three. First, we

want to characterize the B1 and B2 structures using ultra-

soft pseudopotential introduced by Vanderbilt [7] and the

generalized gradient approximation (GGA) [8]. Second, we

will determine the thermodynamic aspects of both struc-

tures through the quasi-harmonic Debye model [9], with

particular attention to the calculations of the quasi-har-

monic Debye model, the volume V, the thermal expansion,

the Gruneisen parameter ratio, (c - c0)/c0, the Debye

temperature H, and heat capacity CV on the pressure P and

temperature T. The last objective of this work is to char-

acterize the phase diagram of B1 ? B2. As known, the

electronic properties and structural phase transformation of

insulators and semiconductors are interesting classical

problems in the solid-state physics. High-pressure research

on structural phase transformations and behavior of mate-

rials under compression, have become quite interesting in

the recent few years as it provides insight into the nature of

the solid-state theories, and determine the values of fun-

damental parameters.

Computational details

The thermal properties and elastic properties calculations

are performed using the pseudopotential plane-wave

method within the framework of the density functional

theory and implemented through the Cambridge Serial

Total Energy Package (CASTEP) Program [10, 11]. This

technique has become widely recognized as the method of

choice for computational solid structural properties inves-

tigations [12]. The thermodynamic properties for ScSb are

calculated by the quasi-harmonic Debye model [9]. The

exchange correlation energy is described in the GGA using

the Perdew–Burke–Ernzerhof (PBE) functional [8]. The Sc

(3d14s2) and Sb (4d105s2p3) states are treated as valence

electrons. Interactions of electrons with ion cores are pre-

sented by the norm-conserving pseudopotential for all

atoms. In all the high precision calculations, the cutoff

energy of the plane-wave basis set is 320 eV for ScSb. The

special points sampling integration over the Brillouin zone

are carried out using the Monkhorst–Pack method with a

8 9 8 9 8 special k-point mesh. The kinetic energy cutoff

and mesh of k points are optimized by performing self-

consistent calculations. The self-consistent is considered to

be converged when the total energy is 10-6 eV/atom.

These parameters are sufficient in leading to well-con-

verged total energy and elastic stiffness coefficients

calculations.

To investigate the thermodynamic properties of ScSb,

we apply the quasi-harmonic Debye model, in which the

nonequilibrium Gibbs function G*(V;P,T) can be written in

the form of [9, 13–16]

G�ðV; P; TÞ ¼ EðVÞ þ PV þ AVibðhðVÞ; TÞ; ð1Þ

where E(V) is the total energy per unit cell, PV corresponds

to the constant hydrostatic pressure condition, and

AVib(H(V);T) is the vibration term, which can be written as

AVibðhðVÞ; TÞ ¼ nKT9h8Tþ 3 lnð1� e�h=TÞ � D

hT

� �� �;

ð2Þ

where n is the number of atoms in the molecule, and the

Debye integral D(H/T) is defined as [9]

DðyÞ ¼ 3

y3

Zy

0

x3

ex � 1dx: ð3Þ

The heat capacity CV and the thermal expansion (a) are

expressed as

CV ¼ 3nK 4Dðh=TÞ � 3h=T

eh=T � 1

� �; ð4Þ

a ¼ cCV

BTV; ð5Þ

where c is the Gruneisen parameter defined as

c ¼ � dlnhðVÞdlnV

: ð6Þ

The Debye temperature, H in Eq. (6), is related to an

average sound velocity, since the vibrations of the solid are

considered as elastic waves in Debye’s theory. For ScSb

crystal, the Debye temperature can be estimated from the

average sound velocity vm, using the following equation

[17]

h ¼ h

kB

3nNAq4pM

� �1=3

vm; ð7Þ

where h is Planck’s constant, kB is Boltzmann’s constant,

NA is Avogadro’s number, M is the molecule mass, q is

the density, and the average sound velocity vm is

approximately given by [18]

6674 J Mater Sci (2012) 47:6673–6678

123

vm ¼1

3

2

v3s

þ 1

v3p

!" #�13

; ð8Þ

where vp and vs are the longitudinal and transverse elastic

wave velocities, respectively, which can be obtained from

Navier’s equation [17]

vp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBS þ

4

3G

� �=q

s; vs ¼

ffiffiffiffiffiffiffiffiffiG=q

p; ð9Þ

where G is the shear modulus and BS is the adiabatic bulk

modulus.

The adiabatic bulk modulus BS is approximated by the

static compressibility [8]

BS � BðVÞ ¼ Vd2EðVÞ

dV2

� �: ð10Þ

Results and discussion

The structural properties are the first step to understand the

material properties with a microscopic point of view. The

calculated equilibrium lattice constants a, zero-pressure

bulk modulus B0 and its pressure derivation B00 from the

Birch–Murnaghan EOS [19] are listed in Table 1, which is

good agreement with other theoretical works [1, 3–6] and

the revealed experimental data [2].

As known, there are two basic methods to obtain the zero-

temperature transition pressure for alloys. The first one is the

slope of the common tangent of the B1 and B2 E–V curves.

The second one is Gibbs free energy, G = H - TS

= E ? PV - TS, i.e., at the phase transition pressure Pt,

Gibbs free energies of B1 and B2 phases attaining the same.

The Gibbs free energy DG (= GB1 - GB2) as a function of

pressure P is illustrated in Fig. 1. The pressure causing

DG approaching to zero is the phase transition pressure (Pt).

The calculated transition pressure (B1 to B2) is

Pt = 42.8 GPa, greater than those predicted by Refs. [1, 3–

6] and close to the experiment data 43 GPa [2]. The calcu-

lated equilibrium lattice constants a, zero-pressure bulk

modulus B0 and its pressure derivation B00 are also consistent

with other theoretical and experimental results available

(Table 1).

The equations of state (EOS) of ScSb in both B1 and B2

phases are obtained using the quasi-harmonic Debye model

[9]. We illustrate the normalized primitive cell volume

V dependences on pressure P in Fig. 2, together with the

experiment data given by Hayashi et al. [2]. Obviously, in

B1 phase, below 20 GPa, our results almost are identical

with Hayashi et al. [2]. Only when pressure goes higher,

the curve of V–P becomes slightly smaller than that of

Hayashi et al. (Fig. 2). We attribute this to applying the

different pseudopotentials. In the investigated pressure the

curve of V–P shows steeper, indicating that ScSb is com-

pressed much more easily at higher pressure. At transition

pressure Pt, our calculation shows that there is volume

change between B1 and B2, which (B1 to B2) decreases

2.79 A3 close to the 2.42 A3 proposed by Maachou et al.

[5], and the structural change from NaCl-type to the CsCl-

type structure occurs with the volume collapse of about

5.7 % in keeping with 5.2 % given by Varshney et al. [6].

The negligible changes of Sc–Sc and Sc–Sb bond distances

with the applied pressures are observed in Fig. 3, together

with the experiment data given by Hayashi et al. [2]. This

result is important since to a first approximation the

vibration frequencies are dependent on bond distances. As

expected experimentally and theoretically, the Sc–Sc and

Sc–Sb bond lengths decrease with pressure. It is shown that

our results are coordinate with the experiment data in the

applied pressures (Fig. 3). We can also note that the Sc–Sc

Table 1 The lattice constants a (A), bulk modulus B0 (GPa) and its pressure derivation B00 (GPa), elastic constants Cij (GPa) of the B1 and B2

structures of ScSb at P = 0 and T = 0, together with the transition pressures Pt (GPa)

This work Other theoretical calculations Experiments

B1 structure

a 5.883 5.887 [5], 5.87 [1], 5.797 [3], 5.793[4]LDA, 5.93 [4]GGA, 5.93 [4]GGA 5.851 [2]

B0 67 70.90 [5], 66.84 [1], 71.3 [3], 78.4 [4]LDA, 64.58 [4]GGA, 58 [6] 58 ± 3 [2]

B00 4.09 3.47 [5], 4.14 [1], 3.65 [3], 3.88 [4]LDA, 3.86 [4]GGA 9.5 ± 0.8 [2]

C11 155GGA, 170LDA 186 [4]LDA, 155 [4]GGA

C12 23.18GGA, 24.2LDA 21 [4]LDA, 20 [4]GGA

C44 23.30GGA, 26.8LDA 26 [4]LDA, 22 [4]GGA, 18 [6]

Pt 42.8 39.78 [5], 40 [1], 16 [3], 35 [4], 31 [6] 43 [2]

B2 structure

a 3.617 3.622 [5], 3.61 [1], 3.61 [3], 3.55 [4]LDA, 3.639 [4]GGA 3.66 [2]

B0 70 69.78 [5], 68.14 [1], 68.14 [3], 83.35 [4]LDA, 67.39 [4]GGA

B00 3.98 3.77 [5], 4.03 [1], 4.03 [3], 3.46 [4]LDA, 3.74 [4]GGA

J Mater Sci (2012) 47:6673–6678 6675

123

bonds shorten slightly faster than the Sc–Sb bonds (Fig. 3),

which shows that there exist a high antiphase boundary

(APB) energy and a fairly large elastic shear anisotropy.

The high APB energy is attributed to the directional

bonding between Sc and Sb electrons.

The thermal expansion parameter a is thought to be

described the alteration in a frequency of the crystal lattice

vibration based on the lattice’s increase or decrease in

volume as a result of temperature changes. It is directly

related to the EOS. We have determined the pressure

dependence of a for both B1 and B2 phase shown in Fig. 4.

It can be seen that the thermal expansion parameter, a,

decreases nonlinearly with the pressure for both B1 and B2

phase. However, the thermal expansion parameter in higher

temperature decreases rapidly with pressure than that in

lower temperature. At Pt, the thermal expansion parame-

ters jump up by 46.1 and 57.7 % at 300 and 2000 k,

respectively.

The variations of the heat capacity CV with pressure

P for the B1 and B2 phase of ScSb are shown in Fig. 5.

They are normalized by (CV - CV0)/CV0, where CV and

CV0 are heat capacity at any pressure P and zero pressure.

The heat capacities decrease almost linearly with the

applied pressures, as shows the fact that the vibration fre-

quency of the particles in ScSb changes with pressure and

temperature. However, at Pt, the higher temperature, the

smaller decrease of CV can be seen.

Anharmonic properties of solids are customarily

described in terms of the Gruneisen parameter c, which can

also be calculated from knowledge of the phonon fre-

quencies as a function of the crystal volume V. We

have determined the pressure dependence of Gruneisen

Fig. 1 The Gibbs free energy DG (= GB1 - GB2) as a function of

pressure P for ScSb

Fig. 2 The volume–pressure diagram of the B1 and B2 structures of

ScSb at various pressures

Fig. 3 The changes of Sc–Sc and Sc–Sb bond distances with the

applied pressures

Fig. 4 The thermal expansion versus temperature and pressure for

ScSb

6676 J Mater Sci (2012) 47:6673–6678

123

parameter c for both B1 and B2, which are shown in Fig. 6.

It is shown that below the phase transition pressure Pt, the

Gruneisen parameter ratios (in B1 phase), (c - c0)/c0,

where c and c0 are Gruneisen parameter at any pressure

P and zero pressure, decrease (nonlinearly) with the pres-

sure, and the higher temperature, the faster decrease of cappears. However, above the phase transition pressure Pt,

the Gruneisen parameter ratios (in B2 phase at 300 and

1800 K), (c - c0)/c0, decrease (nearly linearly) almost in

the same slope with the pressure. At Pt, the ratios of

(c - c0)/c0, jump up by 0.74 % and jump down by

15.53 % at the temperature 300 and 1800 K, respectively.

The pressure dependence of the calculated Debye tem-

perature hD is shown in Fig. 7. For both B1 and B2, hD

increases (nonlinearly in B1 and almost linearly in B2)

with the pressure. At zero pressure, hD is found to be

279.8 K which is about 28.5 % greater than the obtained

for constituent atom Sb (200 K). When the pressure is near

Pt the Debye temperature is higher in B2 phase than that in

B1 phase.

To predict the phase diagram at high temperatures and

pressures, we employ quasi-harmonic Debye model [8],

which has been successfully applied to investigate the

thermodynamic properties of PtN2 [14], TiAl [15], NiAl

[16], MgTe [20], ZrC [21], and so on. The pressure

dependence of the Gibbs free energy difference between

B1 and B2 structures (DG = GB1 - GB2) of ScSb at dif-

ferent temperatures is displayed in Fig. 8. At 0 K (not

shown in Fig. 8), the calculated phase transition pressure

(B1 to B2) is 42.8 GPa, close to the experimental result

Fig. 5 The variations of heat capacity with pressure P. They are

normalized by (CV - CV0)/CV0, where CV and CV0 are heat capacity

at any pressure P and zero pressure at the temperatures of 300 and

1800 K, respectively

Fig. 6 The variations of the Gruneisen parameter ratios with pressure

P. They are normalized by (c - c0)/c0, where c and c0 are Gruneisen

parameter at any pressure P and zero pressure at the temperatures of

300 and 1800 K, respectively

Fig. 7 Variation of Debye temperature (hD) with pressure

Fig. 8 Calculated Gibbs energy difference between B1 and B2 phase

of ScSb (DG = GB1 - GB2) as a function of pressure at different

temperatures

J Mater Sci (2012) 47:6673–6678 6677

123

43 GPa [2].The B1 crystal phase is thermodynamically and

mechanically stable at zero pressure. Conversely, the B2

phase is unstable. As pressure increases, beyond the phase

transition pressure (Pt), the B2 system becomes mechani-

cally and thermodynamically stable while the B1 phase

remains unstable up to the greatest pressure studied. The

increase of temperature results in an increase of DG. At

400 K, DG is already negative at low pressures and, thus,

the B1 structure appears in the calculated phase diagram.

At the same time, DG increases under compression and

eventually becomes positive indicating a stabilization of

the B2 structure, and at higher temperatures the transition

pressure of the B1 ? B2 decreases even further.

Conclusions

We have investigated the hydrostatic pressure-induced

transition phases of ScSb from the B1 to B2 phase by the

ab initio plane-wave pseudopotential density functional

theory method using package CASTEP. It is found that the

transitional phase occurs at 42.8 GPa from the B1 to B2

phase according to the usual condition of equal enthalpy.

Through the quasi-harmonic Debye model, we have suc-

cessfully obtained the dependences of the volume V, the

thermal expansion parameter a, the Gruneisen parameter

ratios, (c - c0)/c0, the Debye temperature H, and heat

capacity CV on the pressure P and temperature T. To pre-

dict phase transition B1 to B2 at high temperatures, we use

the quasi-harmonic Debye model and obtain the phase

diagram of ScSb. More experiments are necessary to fur-

ther make clear the structural sequence of ScSb.

Acknowledgements This project was supported by the National

Natural Science Foundation of China under Grant No. 40804034, and

by the Natural Science Foundation of the Education Department of

Henan province of China under Grant No. 2009B590001 and by

Henan Science and Technology Agency of China under Grant No.

092102210314 and by Open Project of State Key Laboratory of

Superhard Materials (Jilin University) No. 201107.

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