Cent. Eur. J. Phys. • 6(2) • 2008 • 223-229
DOI: 10.2478/s11534-008-0011-7
Central European Journal of Physics
Structural and elastic properties of barium
chalcogenides (BaX, X=O, Se, Te) under high
pressure
Research Article
Purvee Bhardwaj1∗ , Sadhna Singh2† , Neeraj Kumar Gaur1
1 Department of Physics Barkatullah University, Bhopal-462026, India
2 Centre for Basic Sciences M. P. Bhoj (open) University, Bhopal-462016, India
Received 10 August 2007; accepted 12 October 2007
Abstract:
In the present paper we have investigated the high-pressure, structural phase transition of Barium chalcogenides (BaO, BaSe and BaTe) using a three-body interaction potential (MTBIP) approach, modified by
incorporating covalency effects. Phase transition pressures are associated with a sudden collapse in
volume. The phase transition pressures and associated volume collapses obtained from TBIP show a
reasonably good agreement with experimental data. Here, the transition pressure, NaCl-CsCl structure
increases with decreasing cation-to-anion radii ratio. In addition, the elastic constants and their combinations with pressure are also reported. It is found that TBP incorporating a covalency effect may predict
the phase transition pressure, the elastic constants and the pressure derivatives of other chalcogenides
as well.
PACS (2008): 62.20.de, 62.20.dq, 62.50.-p
Keywords:
structural phase transition • Gibbs free energy • volume collapse • phase transition pressure • three-body
interaction
© Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1.
Introduction
High-pressure research on structural phase transformations and the behavior of materials under compression,
based on their calculations or measurements, have become
quite interesting in recent years, as it provides insight
into the nature of solid-state theories, and determines the
∗
†
E-mail: [email protected]
E-mail: [email protected]
values of fundamental parameters. An example can be
found in the III-V semiconducting compounds, where the
structural optical and electrical properties at high pressure have been extensively studied. The alkaline earth
chalcogenides (AX: A=Be, Mg, Ca, Sr, Ba; X= O, S, Se,
Te) form a very important closed-shell ionic system, crystallizing in an NaCl-type structure at room temperature
and pressure, except for the MgTe and beryllium chalcogenides [1, 2]. These compounds are technologically important materials, having many applications ranging from
catalysis to microelectronics. They also have applications
in the area of luminescent devices [3]. Among these com-
223
Unauthenticated
Download Date | 6/17/17 7:46 AM
Structural and elastic properties of barium chalcogenides (BaX, X=O, Se, Te) under high pressure
pounds, barium chalcogenides (BaX, X= O, Se, Te) exhibit
a structural phase transition, even with the application of
moderate pressures of 85 GPa, 6 GPa, and 5 GPa respectively. These compounds share similar band structures
and, in turn have similar physical properties. In order to
understand the band structure of these compounds and
their rate of change with pressure, we examine barium
chalcogenides under pressure from a theoretical point of
view. Apart from this, these compounds exhibit interesting phenomena of metallization with the application of
higher pressure. Metallization is due to band broadening
with increasing pressure and subsequent overlap of the
filled valence band (p-like valence band of the chalcogen
atom) and the conduction band (d-like conduction band
of the cation). Among these compounds, BaTe has the
lowest band-overlap pressure as a result of its smaller
bulk moduli and band gap. To understand some of the
physical properties of these compounds, a detailed description of the electronic structure of these compounds is
needed. Under normal conditions, these compounds crystallize in rock-salt structure and show a different phase
at high pressure. The new phase often shows different physical and chemical properties. At elevated pressure these compounds undergo pressure-induced, structural phase transition from six-fold coordinated NaCl to
a more closed eight-fold coordinated CsCl structure [4–6].
Recently, many efforts were made to interpret the experimental results by using a variety of theoretical models
[7, 8].
The earlier theoretical studies of B1-B2 transitions by
Rao [6], Sims et al. [5], and Froyen and Cohen [9] were
mainly based on the two-body potential mainly. Froyen
and Cohen [9] successfully studied phase-transition phenomena in some alkali chlorides using the pseudopotential, total-energy (PTE) method with minor disagreements in results. They remarked that results could
be improved by including the effect of non-rigidity of ions
in the model. Sims et al. [5] have systematically studied
the thermodynamics and mechanism of the B1-B2 transition in alkali halides and alkaline earth oxides. They
found larger differences with measured values in transition pressures and activation energies. They concluded
that possible reasons for disagreement include the failure
of the two-body potential model.
Motivated by these results and the remarks of Sims et al.
[5] and Froyen and Cohen [9] for incorporating the chargetransfer mechanism, we thought to incorporate the charge
transfer through three-body interactions. It arises from the
deformation of electron shells. This three-body potential
model had been used earlier for the successful predictions
of phase transitions and high-pressure elastic behaviour
of ionic II-VI and III-V compound semiconductors [10] and
divalent metal oxides [11].
We have studied phase-transition phenomena in ionic
compounds using the three-body potential model (TBIP),
which includes long range Coulombic, three-body interactions, and vander Waals and short-range overlap repulsive
interactions. It is felt that this potential model is suitable
for ionic solids but is insufficient to predict the phase transition properties of partially covalent compounds correctly.
So, for partially covalent compounds, we need a potential
model that also takes care of the covalent interaction. To
fulfill this need, we have included the effect of covalency
in the three-body potential along the lines of Motida [12]
and applied it to study the phase transition and elastic
properties of barium chalcogenides.
2. Potential model and method of
calculations
Application of pressure directly results in compression,
leading to increased charge transfer (or the three-body
interaction effect [13]), due to the deformation of the overlapping electron shell of the adjacent ions (or non-rigidity
of ions) in solids.
Also we have considered zero-point energy effects, which
is the lowest possible energy that the compound may possess, and is the ground-state energy of the compound. The
energy of the compound is ε = (hν)/{ehν/kt − 1} + (hν)/2,
where ν, h, t, and k are the frequency, Planck constant,
temperature and Boltzmann constant of the compound. It
is clear from the above expression that even at absolute
zero the energy of the compound cannot be zero but at
least hν/2. This term shows a small effect in Gibbs free
energy, but cannot be ignored completely. Hence there
arises a need to include the zero-point energy term in
TBIP for better agreement with experimental approaches.
These effects have been incorporated in the Gibbs free
energy (G = U + PV − T S) as a function of pressure
and three-body interactions (TBI) [5], which are the most
dominant among the many-body interactions. Here, U is
the internal energy of the system, equivalent to the lattice
energy at temperature near zero, and S is the entropy. At
temperature T = 0K and pressure P, the Gibbs free energies for rock salt (B1, real) and CsCl (B2, hypothetical)
structures are given by
GB1 (r) = UB1 (r) + PVB1 (r),
(1)
GB2 (r 0 ) = UB2 (r 0 ) + PVB2 (r 0 ),
(2)
with VB1 (=2.00r3 ) and VB2 (=1.54r’3 ) as unit cell volumes
for B1 and B2 phases respectively. The first terms in (1)
224
Unauthenticated
Download Date | 6/17/17 7:46 AM
Purvee Bhardwaj, Sadhna Singh, Neeraj Kumar Gaur
and (2) are lattice energies for B1 and B2 structures, and
they are expressed as
(12αm ze2 fm (r))
C
D
−αm z 2 e2
−
− 6 + 8 + 6bβij exp[(ri + rj − r)/ρ]
r
r
r
r
1/2
+6bβii exp[(2ri − 1.414r)/ρ] + 6bβjj exp[(2rj − 1.414r)/ρ] + (0.5)h ω2 B
UB1 (r) =
(3)
1
0
(16αm0 ze2 fm (r 0 ))
D0
−αm0 z 2 e2
C
−
−
+
+ 8bβij exp[( ri + rj − r 0 )/ρ]
r0
r0
r 06
r 08
1/2
+3bβii exp[(2ri − 1.154r 0 )/ρ] + 3bβjj exp[(2rj − 1.154r 0 )/ρ] + (0.5)h ω2 B
UB2 (r 0 ) =
2
With αm and α’m as the Madelung constants for NaCl and
CsCl structures respectively. C (C 0 ) and D(D 0 ) are the
overall van der Waal coefficients of B1 (B2) phases, and
βij (i, j = 1, 2) are the Pauling coefficients. Ze is the
ionic charge, b (ρ) are the hardness (range) parameters,
r(r 0 ) are the nearest-neighbour separations for the NaCl
(CsCl) structure fm (r) is the modified three-body force parameter that includes the covalency effect with three-body
interactions, and ri (rj ) are the ionic radii of ions i (j).
The term hω2 i1/2 is the mean square frequency related to
the Debye temperature (θD ) by
C12 = (e2 /4a4 ) 0.226Z (Z + 12fm (r)) − B1
+ (A2 − 5B2 )/4 + 9.3204zafm0 (r)
(4)
(6)
C44 = (e2 /4a4 ) 2.556Z (Z + 12fm (r)) − B1 + (A2 + 3B2 )/4
(7)
The expression of elastic constants, vander Waals coefficients, and pressure derivatives used by us are reported
elsewhere [15–17].
The expressions for pressure derivatives of second-order
elastic constants (SOEC’s) are as follows:
dk
= −(3Ω)−1 13.980Z (Z + 12fm (r)) + C1 − 3A1 + C2
dp
0
− 3A2 − 167.7648zafm (r) + 41.9420za2 fm” (r)]
(8)
hω2 i1/2 = kθD /h,
Here, θD can be expressed as [14]
1/2
θD = (h/k) (5rBT )/µ ,
with BT and µ as the bulk modulus and reduced mass of
the compounds.
These lattice energies consist of the long-range Coulomb
energy (first term), three-body interactions corresponding
to the nearest-neighbour separation r(r 0 ) (second term),
the vdW (van der Waal) interaction (third term), the energy
due to the overlap repulsion represented by a Hafemeister
and Flygare (HF) type of potential extended to the second
neighbour ions (fourth, fifth, and sixth terms), and the zeropoint effect (last term).
The modified model potential TBP described above for the
NaCl and CsCl phases contains three model parameters
namely the range, hardness and modified three-body force
parameters (b, ρ, fm (r)). The values of elastic constants:
K = 13 (C11 + 2C12 ) , S = 12 (C11 − C12 )
and
Ω = −2.330Z (Z + 12fm (r)) + A1 + A2 + 21.9612zafm0 (r)
C11 = (e2 /4a4 ) − 5.112Z (Z + 12fm (r)) + A1
+ (A2 + B2 )/2 + 9.3204zafm0 (r)
The values of Ai , Bi , and Ci have been evaluated from the
knowledge of b and ρ and the vdw coefficients. We have
added the effect of covalency with three-body interaction
(5)
ds
= −(2Ω)−1 23.682Z (Z + 12fm (r)) + C1 + (C2 + 6A2
dp
− 6B2 )/4 − 50.0752zafm0 (r) + 13.9808za2 fm” (r)
(9)
dC44
= −(Ω)−1 − 11.389Z (Z + 12fm (r)) + A1 − 3B1
dp
C2 + 2A2 − 10B2
+
+ 44.6528Z afm0 (r)
(10)
4
225
Unauthenticated
Download Date | 6/17/17 7:46 AM
Structural and elastic properties of barium chalcogenides (BaX, X=O, Se, Te) under high pressure
on the lines of Motida [12]. Now the modified three-body
parameter fm (r) becomes
fm (r) = fT BI (r) + fcov (r)
fcov (r) =
(11)
2
4Vspσ
e2
rEg3
2
Vspσ
1 − e∗s
(2α − 1)e2
=
, Eg = E − I
2
Eg
12
r
Vspσ is the transfer matrix between the outermost P orbital of the anion and the lowest excited state of the
cation. Eg is the transfer energy of the electron from anion to cation. Denoting the static and optical dielectric
constants ε0 and ε respectively and the transverse optical
phonon frequency at zone centre by ωt , es is represented
as
9µωt2 (ε0 − ε∞ )
=
4πN (ε∞ + 2)2
(e∗s )2
9vµω02 (ε0 − ε∞ )
=
2
(e)
4πe2 (ε∞ + 2)2
(e∗s )2
Table 1.
Input data and model parameters of BaO, BaSe and BaTe.
Input parameters
BaO
BaSe
BaTe
ri [Å]
1.43 [19]
1.43 [19]
1.43 [19]
ri [Å]
1.40 [19]
1.91 [19]
2.11 [19]
r [Å]
2.761 [19] 3.30 [19]
U [KJ/mol]
3832 [19] 2611 [19] 2473 [19]
3.495 [19]
Model parameters
b [10−12 ergs]
02.01
0.982
1.22
ρ [10−8 ]
0.195
0.235
0.225
fm (r)
-0.0301
-0.003
-0.004
ε0
35.6 [22]
11.499 [6] 14.49 [6]
ε∞
3.92 [22]
5.12 [6]
5.95 [6]
ωt [1013 S−1 ]
2.72 [22]
3.09 [6]
2.37 [6]
NaCl (B1 ) and CsCl (B2 ) phases respectively. We have
evaluated the corresponding Gibbs free energies GB1 (r)
and GB2 (r 0 ) and their respective differences ∆G= GB1 (r)
- GB2 (r 0 ), which we have plotted against pressure (P) in
Figs. 1(a)- 3(a) for BaO, BaSe, and BaTe respectively.
where ν denotes the unit cell volume 2r 3 , r is the equilibrium value of the separation of the nearest neighbouring
ions, ε0 the static dielectric constants, µ the reduced mass
of the ions, and ω0 the infrared dispersion frequency.
The values of derivatives of fm (r) are
fm (r) = f0 e−r/ρ + fcov (r)
where various symbols have their usual meaning described
elsewhere [12, 18].
3.
Results and discussion
The three-body potential described in the previous section for NaCl (B1 ) and CsCl (B2 ) structures contains three
model parameters [b, ρ, fm (r)], namely hardness, range,
and the modified three-body interaction parameter. These
parameters are calculated using the lattice-energy equilibrium condition and overlap integrals incorporating the
covalency effect [12]. The input crystal data and model parameters [b, ρ, fm (r)], for barium chalcogenides are given
in Table 1. Using the values of model parameters, we
have computed the phase-transition pressures and volume
collapses for the NaCl (B1) and CsCl (B2) phase at 0 K.
For this purpose, we have followed the technique of minimization of UB1 (r) and UB2 (r 0 ) at different pressures in
order to obtain the inter-ionic separation r and r 0 for the
Figure 1.
Variation of ∆G [KJ/mol] with pressure (a) and of volume
change VP /V0 with pressure (b) for BaO.
The pressure at which ∆G becomes zero [18] is called the
phase-transition pressure Pt . The variation of ∆G and relative volumes V(p) /V(0) with pressure for BaO, BaSe and
BaTe presented in Figs. 1(a)- 3(a) show that ∆G tends
to zero at the phase transition 84 GPa, 5.4 GPa and 5.5
GPa for BaO, BaSe and BaTe respectively. The values
of phase transition pressure for BaSe and BaTe are in
good agreement with the available experimental data [6].
226
Unauthenticated
Download Date | 6/17/17 7:46 AM
Purvee Bhardwaj, Sadhna Singh, Neeraj Kumar Gaur
Figure 2.
Variation of ∆G (KJ/mol) with pressure (a) and of volume
change VP /V0 with pressure (b) for BaSe.
Although the transition pressure of BaO is found to be
more than that of experimentally reported values [3], it is
a better match with available theoretical data obtained
by Jog et al. [11]. It is clear from Table 2 and Figs.
Table 2.
Phase transition Pressure and volume collapse [%] of BaO,
BaSe and BaTe.
Crystal
Phase transition Volume
pressure [GPa]
change [%]
BaO (present)
(Others)
(Exp.)
84
85 [3, 11]
13 [3, 11]
3.3
2.89 [3, 11]
14-16
[3, 11]
BaSe (present)
(Others)
(Exp.)
5.4
12.6
6.0 [6]
12.20
11-14 [6]
BaTe (present)
(MEG)
5.5
5.0 [6]
10.24
11.14 [6]
1(b)- 3(b)) that our calculated volume collapses ∆V(p) /V(0)
from our modified TBP for BaO (2.89%), BaSe (12.20%),
and BaTe (10.24%) are closer than the other studies to
experimental values. Also, we have computed the secondorder elastic constants (SOEC’s) and pressure derivatives
of Barium chalcogenides and reported them in Table 3. It
is clear from Table 3 that our calculated values of SOECs
of Barium chalcogenides are in good agreement with the
experimental data and better than those obtained by others. The calculated pressure derivatives of SOECs could
not be compared due to lack of experimental data on them,
Figure 3.
Table 3.
Variation of ∆G (KJ/mol) with pressure (a) and of volume
change VP /V0 with pressure (b) for BaTe.
Elastic constants and pressure derivatives of elastic modulii of Barium chalcogenides.
Properties BaO
BaSe
C11 [GPa]
89.32
77.87
present
86.8 [20, 21] 66.1 [20, 21] Others
Expt.
112
150 [11]
126 [11]
BaTe
Ref.
C12 [GPa] 45
55 [11]
36 [11]
4.99
3.11
present
2.81 [20, 21] 2.24 [20, 21] Others
Expt.
C44 [GPa] 28
34 [11]
3.86
3.95
present
2.66 [20, 21] 2.41 [20, 21] Others
Expt.
dK/dP
5.88
6.080
present
3.823
3.934 [11] 6.56 [20, 21] 7.00 [20, 21] Others
dS/dP
0.998
6.23
1.019 [11] -
6.90
-
present
Others
dC44 /dP
-0.05
-0.62
-0.02 [11] -
-0.741
-
present
Others
but the magnitude of our SOEC pressure derivatives are
of the same order as reported by others [6].
Figs. 4,5 and 6 show the results on transition volume with
cation radius, volume percent change with cation radius
and transition pressure with cation-anion radius ratio for
barium chalcogenides. They have been compared with the
chalcogenides of Eu, Ca, Sr and Sm, as they belong to the
same family (B1 structure). They all transform from a B1
to B2 structure at high pressure. To understand how the
behaviour of barium chalcogenides are similar or different from other members of the family, we have compared
227
Unauthenticated
Download Date | 6/17/17 7:46 AM
Structural and elastic properties of barium chalcogenides (BaX, X=O, Se, Te) under high pressure
Figure 4.
Variations of transition volume with cation radius for Barium chalcogenides. The solid circles, solid triangles and
solid squares are represent the present, others and experimental values.
Figure 6.
Variation of transition pressure in Barium chalcogenides
as a function of cation to anion radius ratio. The solid
circles, solid triangles and solid squares are represent
the present, others and experimental values.
ume transition and volume change in BaO. Figs. 4 and 5
show that the transition volume and volume change of the
B1 phase at the transition decreases with the size of the
cation. Also we have displayed the transition pressure as
a function of radius ratio (rc /ra ) for barium chalcogenides
in Fig. 6. Our TBP model has yielded results in better
general agreement with the experimental data than others [6] have. The transition volumes and percent change
in volume, along with the experimental data [3] for the B1B2 transition in barium chalcogenides are given in Figs.
4 and 5.
Figure 5.
Variation of volume change (%) ∆V/VT with cation radius
at B1-B2 transition for Barium chalcogenides. The solid
circles, solid triangles and solid squares are represent
the present, others and experimental values.
them. We have shown the transition volume and volume
change (V /VT %) as a function of cation radius for barium
chalcogenides in Figs. 4 and 5 respectively. Inspection of
Figs. 4 and 5 show that the values of transition volume
and volume percent change in BaSe and BaTe are closer
to the experimental data and better than those obtained
by B.S. Rao and S.P. Sanyal [6]. But there is poor agreement between experimental and this study’s values of vol-
For judging shear instability near the phase-transition
pressure, we have computed elastic constants C11 and C12
using a modified three-body potential model and plotted
(C11 − C12 )/2 with increasing pressures in Fig. 7 for BaO,
BaSe, and BaTe. Shear modulii (C11 − C12 )/2 decrease
with pressure, but they do not reach zero at the transition
pressure. This feature is consistent with the requisites of
first-order phase transition. The decrease of (C11 − C12 )/2
with increasing pressure shows that the resistance of these
compounds to any shear decreases as the phase transition
is approached. The values of (C11 − C12 )/2 show a change
at the transition pressure as they switch over to the CsCl
phase.
An overall assessment shows that, in general our values are close to experimental data and they match better
than other theoretical data. The success achieved in the
present investigation can be ascribed to the inclusion of
the charge transfer (or three-body). The covalency effect also seems very important at high pressure when the
228
Unauthenticated
Download Date | 6/17/17 7:46 AM
Purvee Bhardwaj, Sadhna Singh, Neeraj Kumar Gaur
financial support to this work.
References
Figure 7.
Variation of Shear modulii (C11 -C12 )/2 as a function of
pressure.
inter-ionic separation decreases considerably, and the coordination number increases.
Finally, we conclude that the modified three-body potential model has yielded somewhat more realistic predictions of the phase transition and high-pressure behavior
of the barium chalcogenides, compared to those achieved
from more sophisticated models based on a microscopic
approach. The inclusion of three-body interactions with
the covalency effect has improved the prediction of phasetransition pressures over that obtained from the two-body
potential and TBI without covalency. The use of a suitable functional form for the three-body force parameter
with covalency fm (r), instead of using it as a structureindependent model parameter or the three body force parameter f(r), might improve the usefulness of the present
model for estimating the actual high-pressure behavior of
these barium chalcogenides.
Acknowledgement
The authors are grateful to the Madhya Pradesh Council of Science and Technology (MPCST), Bhopal, for the
[1] M. Dadsetani, A. Pourghazi, Phys. Rev. B 73, 195102
(2006)
[2] R. Khenata et al., Physica B 371, 12 (2006)
[3] H.G. Zimmer., H. Winze, K. Syassen, Phys. Rev. B 32,
4066 (1985)
[4] A. Chakrabarti, Phys. Rev. B 62, 1806 (2000)
[5] C.E. Sims, G.D. Barrera, N. L Allan, Phys. Rev. B 57,
11164 (1998)
[6] B.S. Rao, S.P Sanyal, Phys. Stat. Sol. (b) 165, 369
(1991)
[7] A.A. Vaal, D.E. Chernov, Journal of Applied Mechanics
and Technical Physics 27, 427 (1986)
[8] F. El Haj Hassan, H. Akbarzadeh, Comput. Mater. Sci.
38, 362 (2006)
[9] S. Froyen, M.L. Cohen, J. Phys. C 19, 2623 (1983)
[10] R.K. Singh, S. Singh, Phys. Rev. B 39, 671 (1989)
[11] K.N Jog, R.K Singh, S.P. Sanyal, Phys. Rev. B 31,
6047 (1985)
[12] K. Motida, J. Phys. Soc. Jpn. 55, 1636 (1986)
[13] R.K. Singh, Phys. Rep. 85, 259 (1982)
[14] R.K. Singh, D.K. Neb, S.P. Sanyal, J. Phys. C 16, 3409
(1983)
[15] U.C. Sharma, Ph.D. Thesis (Ph.D) (Agra University,
1985)
[16] W. Cochran, Rev. Solid State Phys. Mater. Sci. 2, 1
(1971)
[17] R.K Singh, S. Singh, Phys. Rev. B 45, 1019 (1992)
[18] S. Singh. R.K. Singh, R. Rai, B.P. Singh, J. Phys. Soc.
Jpn. 68, 1269 (1999)
[19] D.R. Lide, (Ed.) CRC Handbook of Physics and Chemistry, 79th edition, 12, 21 (CRC press, Boca - Raton
– Boston – London – New York – Washington, 19981999)
[20] K.G. Straub, A.W. Harrison, Phys. Rev. B 39, 10325
(1989)
[21] Su-Huai Wei, Phys. Rev. Lett. 55, 1200 (1985)
[22] R.K. Singh, S.P. Sanyal, Physica B 132, 201 (1985)
229
Unauthenticated
Download Date | 6/17/17 7:46 AM