A study of the phase transitions, electronic structures and thermodynamic properties of Mg2X (X = Ge, Si and Sn) under high pressure

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Original Article

A study of the phase transitions, electronic structures and

thermodynamic properties of Mg

₂

X (X

¼

Ge, Si and Sn) under high

pressure

M. Guezlane

a

, H. Baaziz

b,*

, Z. Chari

fi

b

, A. Belgacem-Bouzida

c

, Y. Djaballah

c
a_{Department of Physics, Faculty of Science, University of Batna, 05000, Batna, Algeria}

b_{Physics Department, Faculty of Science, University of M'sila, 28000, M'sila, Algeria}

c_{Laboratoire d'}_{etude Physico-Chimique des Mat}_{eriaux, D}_{epartement de Physique, Facult}_{e des Sciences, Universit}_{e de Batna, Rue Chahid Boukhlouf, 05000,}

Batna, Algeria

a r t i c l e i n f o

Article history:

Received 19 November 2016
Received in revised form
21 January 2017
Accepted 26 January 2017
Available online 6 February 2017

Keywords:

DFT
FP-LAPW
EV-GGA

Phase transitions
Thermodynamic

a b s t r a c t

In this work, we theoretically investigate phase transitions, electronic structures and thermodynamic
properties of Mg2X (X¼Ge, Si and Sn) under high pressures. To reach this goal, the total energy has
been calculated by using the full-potential linearized augmented plane wave (FP-LAPW) method with
generalized gradient approximation (GGA), local density approximation (LDA) and EngeleVosko
approximation (EV-GGA), which are based on the exchange-correlation energy optimization. The fully
relaxed structure parameters of Mg2X compounds are in good agreement with the available
experi-mental data. Our results demonstrate that the Mg2X compounds undergo two pressure-induced phase
transitions. The first one is from the cubic antifluorite (Fm3m) structure to the orthorhombic
anticotunnite (Pnma) structure in the pressure range of 3.77e8.78 GPa (GGA) and 4.88e8.16 GPa
(LDA). The second transition is from the orthorhombic anticotunnite structure to the hexagonal Ni2
In-type (P63mmc) structure in the pressure range of 10.41e29.77 GPa (GGA) and 8.89e63.45 GPa (LDA).
All the structural parameters of the high pressure phases are analyzed in detail. Only a small difference
in the structural parameters is observed at high pressures between the calculated and experimental
results. The electronic and thermodynamic properties are also analyzed and discussed. The
estab-lishment of the metallic state of the Mg2X (X ¼ Ge, Si and Sn) compounds at high pressure is
confirmed.

©2017 The Authors. Publishing services by Elsevier B.V. on behalf of Vietnam National University, Hanoi.
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1. Introduction

Among the silicides, Mg2Si is the only possible stoichiometric
compound in the MgeSi alloy as well as Mg2Sn and Mg2Ge. These
compounds have attracted much attention in the last few years due
to their important properties. Their relatively high melting points

(1358K [1], 1030 K[1]and 1390 K[2]for Mg2Si, Mg2Sn and
Mg2Ge, respectively) and high electrical conductivity make them
very useful for high thermoelectric material applications in the
temperature range of 500e800 K[3,4]. The Mg2X (X¼Ge, Si and Sn)
compounds as lightweight materials with high specific strengths
and high specific elastics modulus[5]were proposed to be suitable

materials for the automotive products and manufacturing
pro-cesses, and due to the narrow energy gaps (Eg ~ 0.3e0.6 eV)[6]they
can be used as an infrared detector in the wavelength range from 1.2
to 1.8

m

m [6]. Finally, the non-toxic properties make them
envi-ronmentally friendly[7]. Under ambient conditions (the pressure
below 0.1 MPa), the Mg2X (X ¼Ge, Si and Sn) compounds are
intermetallic with low densities (<2 g/cm3)[8]and crystallize in a
face-centered cubic lattice. They possess the antifluorite (Fm3m)
CaF2type structure[9,10], which is a very interesting type of
sem-iconducting materials forming the simplest metalesemiconductor
hybrid material. Some theoretical and experimental studies have
been conducted to understand the physical properties of Mg2X
(X¼Ge, Si and Sn) in the last few years. The structure and electronic
properties of these semiconductors were reported by several
groups [11e16] with different methods. Some thermodynamic
properties have also been studied[12,17,18].

*Corresponding author. Fax:ỵ213 35556453.

E-mail address:(H. Baaziz).

Peer review under responsibility of Vietnam National University, Hanoi.

Contents lists available atScienceDirect

Journal of Science: Advanced Materials and Devices

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The exploitation of the physical properties of any compound
requires focusing on the relationship between the pressure and the
structure. In fact, the study of the material structure under
compression is a rapidly developing field and is receiving
increasing attention [19]. In 1986, Mao et al. [20] found
experi-mentally by the energy dispersive synchrotron X-ray diffraction
(EDXD) that Mg2Si undergoes a phase transition from the cubic
antifluorite structure to the anticotunnite structure under
pres-sures above 7.5 GPa at room temperature. Recently, Hao et al.[21],
have reinvestigated the structural behavior of this semiconductor
under pressures up to 41.3 GPa. They obtained twofirst order phase
transitions. Thefirst transition occurs at pressures of about 7.5 GPa,
at which the cubic antifluorite (Fm3m) structure changes to the
orthorhombic anticotunnite (Pnma) structure. The second one
oc-curs at higher pressures (of about 21.3 GPa), at which the
com-pound favors the hexagonal Ni2In-type P63mmc structure. Due to
the absence of similar experimental results for the Mg2Sn and
Mg2Ge compounds, Yu et al.[16]have predicted the same phase
transition using the plane-wave pseudo-potential density
func-tional theory method. Looking for the most stable structure of
Mg2X (X¼Ge, Si and Sn), several other computational methods
have been adopted. Most of these calculations are limited to zero
pressure, while the appliactions of the compound are often subject

to a higher pressure above ambient. In literature, however, only a
few results under high pressure in the theoretical works have been
found. Firstly, Kalarasse et al.[22]reportedfirst-principles’studies
of the pressure effect using the full-potential linearized augmented
plane-wave method limited to the ambient structure and without
exceeding 5 GPa. Secondly, Benhai Yu et al. [23] obtained the
structural, electronic, elastic and thermodynamic properties of
magnesium silicide successfully using the first-principles
plane-wave pseudo-potential (PW-PP) method in combination with the
quasi-harmonic Debye model but also without exceeding the 8 GPa.
Finally, Yu et al.[15,16]investigated the phase transitions of Mg2X
(X ¼Ge, Si and Sn) and Huan et al. [14] for Mg2Si under high
pressures using thefirst-principles plane-wave method within the
pseudo-potential and generalized gradient approximations (GGA).
In this work, we have calculated the pressure and temperature
dependence of the thermodynamic properties of the MgeX
(X_¼Ge, Si and Sn) alloys with GIBBS2 program using the WIEN2K
data within the framework of the quasiharmonic approximation. In
addition, the structural and electronic properties of their
stochio-metric compounds Mg2X (X¼Ge, Si and Sn) have been investigated
by using thefirst principle calculations based on density functional
theory (DFT)[24,25]within the full-potential linearized augmented
plane wave (FP-LAPW) method.

2. Computational details

The Mg2X (X¼Ge, Si and Sn) compounds crystallize in a cubic
antifluorite structure at ambient conditions, the Mg and X (X¼Ge,
Si and Sn) atoms occupy the 8c (0.25, 0.25, 0.25) and the 4a (0, 0, 0)
Wyckoff sites, respectively. At high pressure, it has been reported

experimentally [21] that Mg2Si undergoes two structural
trans-formations,firstly to the orthorhombic and then to the hexagonal
structures with a remarkable difference in the volume collapse
between thefirst and second transitions.

The calculations have been performed using the FP-LAPW as
implemented in WIEN2K[26]code based on the very powerful
prediction method for the new materials properties (DFT). In this
FP-LAPW method, the unit cell of the three structures is
parti-tioned into non-overlapping muffin-tin spheres around the
atomic sites and an interstitial region. We used the generalized
gradient approximation (GGA [27]) and the local density
approximation (LDA[25])eby Perdew et al-exchange-correlation

potential to treat the electroneelectron interaction. In addition,
we have applied the EngeleVosko (EV-GGA[28]) scheme which
proposes better electronic properties. In order to achieve energy
eigenvalues convergence, the wave functions in the interstitial
region have been expanded in plane waves with a cut off of
Kmax ¼ 9/Rmt, where Rmt denotes the smallest atomic sphere
radius and Kmaxgives the magnitude of the largest k-vector in the
plane wave expansion. The Rmtis taken to be 2.1e2 atomic units
(a.u.) for Mg and X (X¼Ge, Si and Sn) for all phases.
Brillouin-zone (BZ) integrations within the self-consistency cycles have
been performed via a tetrahedron method, using 35kpoints for
both phases in the IBZ. The self-consistent iterations have been
performed until the convergence in the energy reached about
104 _Ry3_{. We have also used our results obtained by the GGA}
approximation for the thermodynamic properties the GIBBS2
[29]program.

3. Results and discussion

3.1. Total energy calculation and high pressure structural
transformation

We have determined the structural properties from the
calcu-lation of the ground state energy as a function of the volume
around the equilibrium. The variations of the energy (E) with
vol-ume (V) in three structures for the three compounds using GGA and
LDA approximation are shown inFig. 1. The calculated structural
parameters from these three structures' types of Mg2X (X¼Ge, Si,
and Sn) are listed with the available experimental data and few
other theoretical results inTable 1. The obtained lattice parameters
of the antifluorite structure using LDA are in excellent agreement
with the experimental data and other theoretical results at 0 GPa,
whereas the calculated parameters using GGA deviate with some
proportions. For the Mg2Si compound in the hexagonal Ni2In-type
(P63mmc) structure our calculated value of c/a is about 1.3 using
both GGA and LDA approximation in 0 GPa, and 1.27 in the
tran-sition pressure, with the same value found in the prediction of the
two other compounds Mg2Sn and Mg2Ge which is close to the other
experimental and theoretical value 1.26, but a more evident
discrepancy can be observed in the cell parameters at high pressure
phases between our calculated results using GGA and LDA for the
three compounds (Table 1). Using the plane-wave pseudo-potential
density functional theory method, Yu et al.[15,16]and Huan et al.
[14]have found an overlapping curve of the EeV plot between the
anticotunnite and Ni2In-type structures for all the Mg2X (X¼Ge, Si
and Sn) compounds, because of a groupesubgroup relation

be-tween the anticotunnite (Pnma) and the Ni2In-type (P63/mmc)
structures. However, this overlap disappears in our results and the
discrepancy became very clear, as can be explained by the higher
precision of the full-potential method instead of the pseudo
po-tential density functional theory method. We notice here that there
is no accurate experimental data regarding the high pressure
structural behavior available for Mg2Sn and Mg2Ge, and the only
prediction results are obtained by Yu et al. [16]. In the present
study, we can determine the actual transition point between the
anticotunnite and the Ni2In-type structures from our EeV curves.
The transition pressures from the antifluorite phase to the
anti-cotunnite phase and further to the Ni2In-type structure phase are
listed inTable 2with the volume reduction for Mg2X (X¼Si, Sn and
Ge) compounds in comparison with the previous calculations and
experimental data. For all Mg2X (X¼Ge, Si and Sn) compounds, we
can notice that the bulk modulus increase with each transition, and
decrease from LDA to GGA in the same phase. This increase can be
attributed to the increase in the bond strength between atoms
under high pressures.

M. Guezlane et al. / Journal of Science: Advanced Materials and Devices 2 (2017) 105e114

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The relation between pressure and volume using LDA
approxi-mation for the different phases of the Mg2X (X¼Ge, Si and Sn)
compounds is shown inFig. 2. The two phase transitions can be
observed with a volume collapse synonym of a discontinuity in the
pressure. These results indicate that these two transitions are
considered to be offirst-order due to the discontinuity of the
vol-ume at each one of them. The Mg2X (X¼Ge, Si and Sn) compounds
undergo two crystallographic transitions, the first one from the

antifluorite to anticotunnite phase, and the second one from
anticotunnite to the Ni2In-type structures phase. For Mg2Si, thefirst
transition occurs at 8.78 GPa (GGA) and 8.16 GPA (LDA) with a
volume collapse of 11.99% (GGA) and 10.85% (LDA). This is very close
to the values of thefirst transition of Mg2Ge with 12.29% (GGA) and

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Table 1

Calculated lattice parameters, bulk modulus, and DOS at EFusing LDA and GGA, in comparison with the available previous calculations and the experimental data for Mg2X

(X¼Ge, Si and Sn) in three phases.

This work Others Exp.

GGA LDA

Mg2Si-AF

a0(Å) 6.369 6.262 6.09a, 6.26e, 6.29f 6.338g, 6.35h

B (GPa) 54.16 56.55 59.2a_{, 58.3}e_{, 56.2}f _46.3

e55.0a
Mg2Si-HEX

0 GPa 22.9 GPa 0 GPa 22 GPa

a0(Å) 4.651 4.38 4.567 4.309 4.162i 4.166b

c0(Å) 6.046 5.57 5.937 5.601 5.25i 5.287b

c0/a0 1.3 1.27 1.3 1.27 1.261i 1.269b

B(GPa) 55.6236 e 59.4477 e 56.07i 163.83b

Mg2Si-AC

0 GPa 8.78 GPa 0 GPa 8.16 GPa

a0(Å) 6.985 6.89 6.862 6.80 6.595i 6.035b

b0(Å) 4.191 4.13 4.117 4.08 3.995i 4.591b

c0(Å) 8.102 7.99 7.960 7.89 7.734i 6.784b

B(GPa) 57.3325 e 60.5048 e 56.48i 102.65b

Mg2Sn-AF

a0(Å) 6.827 6.675 6.694j 6.759d,6.765b, 6.762e, 6.761f

B(GPa) 37.5718 43.5460 44.74j _41.2k

Mg2Sn-HEX

0 GPa 10.41 GPa 0 GPa 8.89 GPa

a0(Å) 4.996 4.826 4.866 4.769 V0(Å)¼66.49j

c0(Å) 6.495 6.129 6.327 6.057

c0/a0 1.3 1.27 1.3 1.27

B(GPa) 38.2066 e 45.7508 e 46.05j

Mg2Sn-AC

0 GPa 3.77 GPa 0 GPa 4.88 GPa

a0(Å) 7.488 7.49 7.325 7.306 V0(Å)¼69.21j

b0(Å) 4.493 4.49 4.395 4.384

c0(Å) 8.687 8.69 8.497 8.475

B(GPa) 46.2965 50.809 e 45.91j

Mg2Ge-AF

a0(Å) 6.431 6.295 6.12a, 6.286e, 6.31f, 6.423c 6.393g,6.378b, 6.393e, 6.445f

B(GPa) 46.2945 52.7659 57.6a_{, 55.9}e_{, 55.1}f _44.0

e54.7a
Mg2Ge-HEX

0 GPa 29.77 GPa 0 GPa 63.45 GPa

a0(Å) 4.730 4.372 4.658 4.067 V0(Å)¼57.45j

c0(Å) 6.148 5.553 6.055 5.165

c0/a0 1.3 1.27 1.3 1.27

B(GPa) 46.5224 53.4534 51.74j

Mg2Ge-AC

0 GPa 7.85 GPa 0 GPa 8.16 GPa

a0(Å) 7.076 6.946 6.919 6.822 V0(Å)¼59.54j

b0(Å) 4.246 4.168 4.151 4.093

c0(Å) 8.208 8.058 8.026 7.913

B(GPa) 49.3302 e 61.3158 e 54.97j

a_{PWPP Ref.}_[11]_.
b_Ref._[21]_.
c _Ref._[30]_.
d _Ref._[31]_.
e_{FP-LAPW Ref.}_[32]_.
f _{PWPP Ref.}_[12]_.
g_Ref._[33]_.
h_Ref._[13]_.
i_Ref._[15]_.
j_Ref._[16]_.
k_Ref._[11]_.

Table 2

Calculated transition pressure and volume collapse using LDA and GGA, in comparison with the available previous calculations and the experimental data for Mg2X (X¼Ge, Si

and Sn) compounds.

Antifluorite to Anticotunnite Anticotunnite to Ni2In-type

Present Work Theory Experiment[21] Present Work Theory Experiment[21]

GGA LDA GGA LDA

Mg2Si PT(GPa) 8.78 8.16 8.38[15] 7.5e10.4 22.9 22 28.84[15] 21.3e37.8
DV (%) 11.99 10.85 7.53[15] ~12 18.61 17.72 3.66[15] ~3.0
Mg2Sn PT(GPa) 3.77 4.88 5.26[16] e 10.41 8.89 18.40[16] e

DV (%) 8.15 8.73 7.43[16] e 15.39 12.10 3.11[16] e

Mg2Ge PT(GPa) 7.85 8.16 8.71[16] e 29.77 63.45 33.28[16] e
DV (%) 12.29 11.43 6.82[16] e 21.19 33.03 3.12[16] e

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the Mg2Si; 21.19% (GGA) and 33.03% (LDA) for Mg2Ge). However,
Mg2Sn has a lower pressure transition (10.41 GPa (GGA) and
8.89 GPa (LDA)) and a little less volume collapse compared with the
two other compounds (15.39% (GGA) and 12.10% (LDA)). Having
compared with the theoretical results[14e16], we have found that
for this prediction transition there is a rapprochement between
22 GPa (GGA), 24 GPa[14]and 28 GPa[15]for the pressure

tran-sition of Mg2Si, 10.41 GPa (GGA) and 18.40 GPa[16]for Mg2Sn, and
29.77 GPa (GGA) and 33.28 GPa[16]for Mg2Ge, with a small
in-crease. The only big difference between these results is the second
phase transition of Mg2Ge at 63.45 GPa (LDA) with our GGA result
of 29.77 GPa. This discrepancy can be attributed to the
non-existence of the hexagonal Ni2In-type (P63mmc) structure phase
for Mg2Ge.

3.2. Band structure and density of states

We have computed the band structure and the total and partial
density of state (DOS) of Mg2X (X¼Ge, Si and Sn) compounds in the
antifluorite (AF), anticotunite (AC) and hexagonal Ni2In-type (HEX)
structures using GGA, LDA and EV-GGA approximations to show
the pressure effects on these properties. It was well known that the
simple form of GGA and LDA is not sufficientlyflexible for
accu-rately reproducing both exchange-correlation energy and its charge
derivative. They usually underestimate the energy gap [34,35].
That's why Engel and Vosko[28]by considering this shortcoming
constructed a new functional form of the GGA (called as EV-GGA),
which can provide a better band splitting and some other

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Fig. 3.Electronic band structures and total density of states (TDOS) of Mg2X (X¼Ge, Si and Sn) compounds calculated using EVGGA.

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results show that the valence electrons are mainly around X,
although there is a little indication of a weak covalent bonding
between Mg and X. With increasing pressure, the valence band
becomes wider and the conduction band penetrates down in the

valence band. Thus, the value of DOS at EF(zero in our case)
in-creases. The principal contribution to DOS near EFcome from
2p-Mg and p-X states for the anticotunite phase. Going further with
the pressure, the bands become wider and the value of DOS at EF
tends to decrease in the hexagonal Ni2In-type phase, this
over-lapping band explains the metallization of the Mg2X compounds.
3.3. Thermodynamic properties

Thermodynamic properties including heat capacity, thermal
conductivity, thermal expansion and the Grüneisen parameter are
fundamental features of materials. They give interesting
informa-tion such as thermodynamic stability, interatomic interacinforma-tions,
anharmonicity of lattice vibrations and the utility of materials for
various applications.

As we have mentioned, Mg2X (X¼Ge, Si and Sn) compounds are
characterized by two phase transitions at high pressure, which can
be explained by the effect of temperature on these two transitions
and generally on the properties of each phase. Therefore, it is very

important to study the thermodynamic properties and the effect of
temperature on some structural parameters of these compounds in
each phase (the heat capacity, the expansion coefficient, the Debye
temperature, the bulk modulus and the relative variation in
vol-ume). We started in Fig. 4 with the effects of temperature and
pressure on the bulk modulus B to get some information about the
resistance to the contraction in each phase by plotting the variation
of B as a function of temperature for three different pressure values
0, 20 and 50 GPa using the GGA approximation. In overall, for low
temperatures between 0 and 100 K the bulk modulus appears

constant especially at high pressure (50 GPa) in the three phases of
Mg2X (X¼Ge, Si and Sn) compounds (a change ofỵ0.27% (for the
Mg2Ge hexagonal phase at 50 Gpa) to 2.81% (for the Mg2Ge
Fig. 3.(continued)

Table 3

Band gap of Mg2X (X¼Ge, Si and Sn) in the antifluorite phase.

GapG-X (eV)

Our work Other results[36]

LDA GGA EV-GGA

Mg2Ge 0.097 0.168 0.701 0.7

Mg2Si 0.116 0.2218 0.676 0.6, 0.57[14]

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hexagonal phase at 0 Gpa)). Above 100 K, bulk modulus decreases
linearly with increasing temperature up to 1000 K but differently
under each pressure. The maximum percentage of changes clearly
seen from these results is about57.85% for Mg2Ge in the
hexag-onal phase structure under 0 GPa of pressure. We note here that
increasing the pressure decreases clearly the influence of
temper-ature on the bulk modulus B. Under zero-presure and at 0 K, the
antifluorite (AF) phase has the smallest bulk modulus. B is lager in
the hexagonal (HEX) phase and then further increases in the
anti-cotunite (AC) phase for all Mg2X (X¼Ge, Si and Sn) compounds.
The order change between the antifluorite and the hexagonal

phases is observed at 390 K and 351 K for the Mg2Si and Mg2Ge
compounds, respectively. Under a pressure of 20 GPa, a change of
order from AF-HEX-AC (with the smallest B value) to HEX-AF-AC
(with the highest B value) can be observed for Mg2Ge at about
784.5 K. For Mg2Si, the change from AF-AC-HEX to AF-HEX-AC
occurs at about 633 K. Only one change from AC-HEX to
AF-AC-HEX is observed under a pressure of 50 GPa for Mg2Ge at 906 K.
Mg2Si always maintains the AF-AC-HEX arrangement under the
pressure of 50 GPa. For Mg2Sn, the order is AF-HEX-AC, HEX-AC-AF
and HEX-AC-AF under pressures of 0, 20 and 50 GPa, respectively
without any temperature effects. However the most noticeable
change under pressure effect is the behavior of the values of bulk
modulus in the antifluorite phase of Mg2Sn which increases with
pressure to exceed the value of the two other phases. The calculated
volumetric thermal expansion coefficients (

a

) are plotted as a
function of temperature inFig. 5under three pressures 0, 20, and
50 GPa. For high pressures (20 and 50 GPa), a very fast expansion
can be clearly detected under 200e300 K, then it becomes very
slight until reaching a saturation value which depends for each
phase on the applied pressure. Unlike these results, the alpha
co-efficients under 0 GPa for Mg2Ge and Mg2Si continue their
increasing value after 300 K, not like thefirst rate but with a clear

evolution. The same behavior for Mg2Sn is observed but just in the
antifluorite phase which causes the only transition observed
be-tween the value of this coefficient under the temperature effect
between the hexagonal and the antifluorite phases at 412 K. The
other remarkable result obtained from this curve is the great effect
of the pressure to this thermal expansion coefficient, which reduces
its value ranges from between 10.5105K1and 27105K1

under 0 GPa to between 3.3105K1and 4.6105K1and
between 1.8 105 K1 and 2.7105 K1 under 20 GPa and
50 GPa respectively, which is a reduction of 60%e88%. Another
remark is regarding the structure which has the biggest value of the
alpha coefficient at 0 GPa, it is always the hexagonal phase
struc-ture except for Mg2Sn which is characterized by the increase of
alpha in the antifluorite phase structure under the temperature
effect at this pressure, which can be reviewed in result of the
constant pressure heat capacity.Figs. 6 and 7show our calculated
constant volume heat capacity Cvand constant pressure heat
ca-pacity Cpas a function of temperature for three different pressure
values 0, 20, and 50 GPa using the GGA approximation. At low
temperature, both Cvand Cpincrease rapidly with temperature till
200 Ke300 K then this increase becomes lower for Cvto reach the
saturation values given by Delong and Petit[37]Cv¼3nNAKBwhich
corresponds to ~75 J/mol.K for Mg2X (X¼Ge, Si and Sn) compounds
in all phases. However, Cpcontinues increasing with temperature
but slowly under 0 GPa with the same notice given for the alpha
coefficient. The results obtained for Debye temperature (

q

D) are
plotted as a function of temperature under three different pressures
in Fig. 8. The parameter

q

D increases clearly with pressure and
decreases very slowly with temperature especially under high
pressure. We also added the variation of Gibbs energy (G) under
temperature and pressure effects inFig. 9, which shows the
tran-sition phase and the weak effect of the temperature relative to the
pressure on this energy (G).

Fig. 4.Temperature and pressure effect on the bulk modulus B for Mg2X (X¼Ge, Si and Sn) compounds.

Fig. 5.Temperature and pressure effects on the volumetric thermal expansion coefficients (a) for Mg2X (X¼Ge, Si and Sn) compounds.

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Fig. 7.Temperature and pressure effects on the constant pressure heat capacity CPfor Mg2X (X¼Ge, Si and Sn) compounds.

Fig. 8.Temperature and pressure effects on the Debye temperatureqDfor Mg2X (X¼Ge, Si and Sn) compounds.
Fig. 6.Temperature and pressure effects on the constant volume heat capacity Cvfor Mg2X (X¼Ge, Si and Sn) compounds.

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4. Conclusion

We have performed the first principle calculations using the
full-potential linearized-augmented plane wave method
(FP-LAPW) to investigate the structural, electronic and thermodynamic
properties of Mg2X (X¼Ge, Si and Sn) compounds for antifluorite,
anticotunite and hexagonal Ni2In type phases. The
exchange-correlation potential has been treated using three different
ap-proximations of LDA, GGA and EV-GGA. The obtained results for
equilibrium unit cell volumes and bulk modulus at zero pressure
are rather close to those reported in the literature. At pressures
below 3 GPa, the Mg2X (X¼Ge, Si and Sn) compounds maintain
their antifluorite structure with different bulk modulus values of
46.52 GPa, 54.16 GPa and 37.57 GPa for Mg2Ge, Mg2Si and Mg2Sn,
respectively. At high pressures, these compounds undergo two
crystallographic phase transitions offirst-order nature to become
the hexagonal structure. The density of state and the band structure
have been calculated by using EV-GGA for the Mg2X compounds in
all three phases, showing the metallic character for the two last
phases, which is in good agreement with the previous calculation.
In addition, we have used GIBBS2 program to introduce the
tem-perature effect in these ab-initio results which allowed us to

calculate the constant volume and pressure heat capacity, Debye
temperature and the Gibbs free energy, as functions of temperature
and pressure.

Acknowledgments

This work is supported by the Algerian University research
project (CNEPRU) under no. D05620140014.

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