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8FIRST-PRINCIPLES INVESTIGATION THE PHASE SEPARATION
IN CabxMg,O ALLOYS
R. Miloua, F. Miloua, Z. Kebbab, N. Benramdane
Laboratoire d'Elaboration et de Caractérisation des Matériaux, Faculté des Sciences de l'Ingénieur, BP 89, Université Djillali LIABES Sidi-Bel-Abbès, Algeria. Tel: 00 213 50 71 61 56; E-mail: [email protected]
Using the full-potential linearized augmented plane wave (FP-LAPW) method in combination with the local density approximation to the exchange-correlation potential, we investigated the ground-state properties and the stability of Ca1-xMgxO mixed oxides. It is found that the structural parameters, i.e. lattice constants and bulk moduli deviate slightly from the linear function of the composition x. We determined the equation of state of the alloys and showed an increasing compressibility function of composition. We expressed the formation energy as an energetic balance between pure structural constraints and quantum chemical effects. Thus, a phase separation over the whole range of concentration is expected. The origin of the miscibility gap has a chemical nature. Also, we performed a thermodynamic study of the stability of the alloys.
Keywords: development and study of material properties to form catalytic layers in fuel cells, new structural materials for renewable energy structures
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Organization(s): Scientist researcher at Laboratoire d'Elaboration et de Caractérisation des Matériaux. Education: D.E.A in electronics at Université des Sciences et Techniques of Lille, France (1986). Doctorate in Physics and materials science at Claude Bernard University- Lyon I, France (1992). Main range of scientific interests: Materials science, surfaces and interfaces.
Some publications (number of publications: 11): F. Miloua et al. Structural and electrical characterzation of Ag-InP(100) interfaces stabilized by antimony // Materials Chemistry and Physics 1993. 33. P. 85.; M. Bouslama, B. Khelifa, F. Miloua et al. Interaction of phosphorus with Indium // Appl. Surf. Sciences. 1992.; F. Miloua et al. Structural and electrical characterizations of Ag-InP(100) interfaces stabilized by antimony // 3eme Rencontre de Physique: ''Sciences des Surfaces des Matériaux''. Oran. 1991.
F. Miloua
Introduction
The alkaline earth oxides CaO and MgO, as well as their mixtures have been in focus both theoretically and experimentally. Because of their interesting physical and chemical properties, they found a wide range of applications ranging from catalysis to microelectronics. They have been used as catalysts for the steam gasification of naphthalene [1], in oxidative coupling reaction of methane [2] and for soot combustion [3]. Recently, Ca1.xMgxO solid solutions have been demonstrated as an alternative dielectrics to SiO2 due to favourable properties such as high dielectric constant, wide bandgap, and notable lattice compatibility with SiC [4]. CaMgO films grown by rf plasma-assisted molecular beam epitaxy and capped with Sc2O3 are promising candidates as surface passivation layers and gate dielectrics on GaN-based high electron mobility transistors (HEMTs) and metal-oxide semiconductor HEMTs (MOS-HEMTs) [5, 6].
Due to the large difference in ionic radius between Mg and Ca, solid-solution CaO-MgO is difficult to synthesize using bulk techniques, i.e. a severe
immiscibility is observed [7]. However, the use of MBE as a film growth technique often allows for the formation of metastable phases. In fact, Ca1-xMg^O ternaries were grown as solid solution films covering the entire compositional range when grown at a remarkably low temperature of 300 °C [8]. Also, using pulsed laser deposition, Nishii et al. have grown metastable solid solution films on ZnO layers [9]. In the present paper we report on the structural and thermodynamic stability of Ca1-xMg^O alloys. We employed the Full-Potential Linearised Augmented Plane Wave (FP-LAPW) method [10] to study the ground-state properties and the formation energy of the alloys. The disordering effect on the formation energy is introduced using a cluster expansion approach. The predicted trends are explained on the basis of a structural-chemical energetic balance. Further, the thermodynamic phase diagram is established and the critical temperature is estimated. The rest of the paper is organized as follows. In section 2, we briefly describe the calculation procedure. Results and discussions are presented in section 3 and the paper is concluded in section 4.
Method of calculation
Our calculations have been made using FP-LAPW approach within the framework of the Density Functional Theory (DFT) [11, 12] as implemented in WIEN2k [13] code. The exchange-correlation contribution to the total energy is described within the Local Density approximation (LDA) [14]. Kohn-Sham wave functions were expanded in terms of spherical harmonic functions inside the non-overlapping muffin-tin spheres surrounding the atomic sites (MT spheres) and in Fourier series in the interstitial regions. Inside the MT spheres of radius RMT, the l-expansion of the wave function were carried out up to lmax = 10 while the charge density was Fourier expanded up to Gmax = 14 (Ryd)12. In order to achieve energy eigenvalues convergence, the wave functions in the interstitial region were expanded in plane waves with a cut-off parameter of Kmax = 8/ftMT for both binary and ternary compounds. RMT values were assumed to be 2.0 a.u. for Mg, 2.2 a.u. for Ca and 1.6 a.u. for O atoms, respectively for all structures. A mesh of 30 special k-points for both binary and ternary compounds was taken in the irreducible wedge of the
Brillouin zone. Both the MT radius and the number of k-points were varied to ensure total energy convergence. The core states that are completely confined inside the corresponding MT spheres were treated fully relativistic, while for the valance states we used the scalar relativistic approach that includes the mass velocity and Darwin s-shift, but omits spin-orbit coupling.
Results and discussions
Structural parameters First, we calculated the structural properties of the binary compounds CaO and MgO in the rock-salt structure. Then, the ordered ternary alloys Ca1-xMgxO (0 < x < 1) were simulated at compositions x = 0.25, 0.5 and 0.75 by substituting cations. For each composition, we carried out a structural optimization by minimizing the total energy with respect to the cell volume and also the atomic positions. The calculated lattice constants and bulk moduli (see Table 1) were obtained by fitting the total energy versus unit cell volume to the Murnaghan's equation of state [15].
Table 1
Calculated lattice constants and bulk moduli, compared to experimental and other theoretical results
Ca1-xMgxO Lattice constants, a (Â) Bulk moduli (GPa)
x This work Exp Other calc This work Exp Other calc
0 4.71 4.81 [16] 4.72 [17] 127 110 [16] 128 [17]
0.25 4.61 135.84
0.5 4.49 144.41
0.75 4.35 157.12
1 4.17 4.213 [18] 4.165 [17] 173.15 160 [18] 171 [17]
In Fig. 1 and Fig. 2 we depicted, respectively the lattice constants and the bulk moduli as a function of the composition x. From Fig. 1, one can see that the calculated lattice parameters deviate slightly from those obtained from the Vegard's law. The differences between themes are less than 5.5 %. The calculated lattice constants follow a decreasing quadratic function of composition x
a(x) = 4.70961 - 0.32582x - 0.21422x2. (1)
In Fig. 2, the composition dependence of the bulk moduli is compared with the results predicted by the linear concentration dependence (LCD). The bulk moduli exhibit a strong deviation from the LCD. Considering the general trend that LDA usually underestimates the lattice constant and overestimates the bulk modulus, our results are in good agreement with the experimental and other calculated values (see Table 1).
-|-1-*-1-*-1-*-1-*-1-*-r
IS 1,3 --
o.o o; o,o o¡s ijo
Composition *
Fig. 1. The calculated lattice constants (solid squares) and lattice constants of ideal mixing solid solutions (doted line) for the five ordered structures Ca1.„MgnQ (n = 0, 1, 2, 3, 4)
R. Miloua, F. Miloua, Z. Kebbab, N. Benramdane. First-principles investigation the phase separation in Ca1.xMgxO alloys
Fig. 2. Calculated bulk moduli (solid squares) and those of ideal mixing (dotted line) as a function of composition
Equation of state The calculated equation of state for the Ca1-xMgxO alloys is plotted in Fig. 3. The relationship of the strain volume V/V0 (where V is the unit cell volume under pressure, V0 is the volume without pressure) under the same pressure P among the different compositions can be written as
VVo )caO <V/V> ), <V/Vo \
Ca3MgO4
CaMg3O4
■■(V/Vo )
■(V/Vo )
CaMgO2
MgO
(2)
OÍS -
Pressure [G Pa]
Fig. 3. Equations of state of the Ca1-xMgxO alloys
Phase separation of the CaMgO alloys
The ordered phases In order to study the stability of CaMgO in the ordered form, we calculated the formation energies of the five (n = 0, 1, 2, 3, 4) structures using the following relation
( ) n f n ^
Eform (n) = ECa1-nMgnO EMgO EcaO , (3)
where E,
Cai-nMg nO
is the total energy of the nth ordered
alloy, EMgO and ECaO are total energies of MgO and CaO binaries, respectively. The results are viewed in Fig. 4 (solid squares) together with those for disordered alloys (solid line).
0.Í0 0,35 0,30
o;s
"¡D
0,15
0,10 0J05 0|00
■ Ordered phase
■ Disordered phase
0Í
0Í
Composition x
The relationship shows that CaO can be compressed more easily than MgO, this trend is observed in the bulk modulus (5CaO < BMgO). The compressibility decreases when the composition x increase.
Fig. 4. The calculated formation energies for ordered structures and disordered alloys
The formation energy can give a measure of the stability of a material. We notice that it is positive for the whole range of composition, this indicates that it is energetically unfavourable for CaO and MgO to mix and form alloys. In order to clarify the physical origin of this behaviour, we decompose Eform into three physically recognizable contributions [19-21] as follows:
Eform EV
i + Ece + ESI
(4)
where Eyu is the hydrostatic "volume deformation" contribution due to the dilatation of MgO and compression of CaO from their equilibrium lattice constants (4.17 A and 4.71 A, respectively) to the common (Vegard-like) lattice constant of the alloy; Ece is the "charge exchange" energy that releases when MgO and CaO, taken at common lattice constant, combine to give Ca1-xMgxO ordered alloy. The internal relaxation effects are not included in this energy. ESR is the "structural relaxation" term due to the full relaxation of cell-internal degrees of freedom. Fig. 5 shows the splitting of the formation energies of the ternary alloys into the three physical contributions; we can deduce the following important remarks:
- The large lattice mismatch (~ 12%) between the binary constituents leads to a large and positive Evu that tends to decrease the stability of CaMgO alloys over the whole range of composition.
- In contrast to EVD, internal relaxation tends to increase the stability of the alloys by taking a negative ESR value. However, ESR is very small, compared to EVD and the sum EVD + ESR remains positive.
<
- The ECE contribution remains positive and is higher than + Esr, so can say that the phase separation has a chemical nature.
kBT / AH 0 = (8x - 4)/[ln x - ln(l - x)].
(8)
Composition x
Fig. 5. Energy contributions to the formation energies of ordered alloys
The disordered phase In this section, the energetic properties of the disordered alloys are calculated using a cluster expansion method. Following the idea of Connolly and Williams [22], the energy of formation of the disordered solid solutions can be written as
Etm (x) = E Pn (xf (x)
n=0
Pn (x) = f 41 xn (1 - x )4-n
(5)
(6)
where E"foim (x) is the energy of formation of each of the
five ordered structures. Pn(x) is a statistical weight representing the probability that the nth short-range ordered structure occurs in the alloy. In Fig. 4 (solid line), the formation energy of the disordered alloys is depicted. It is lower than that of the ordered structures.
Thermodynamic properties The thermodynamic properties of Ca1-xMgxO are calculated by estimating the solubility limit. Following the approach of Neugebauer and Van de Walle [23], we use the calculated formation energy to construct a lower limit AHmin(x) for each composition
AH mm (x) = 4 x(l - x)AHc
(7)
where AH0 = 0.1351863 eV (AHmin (x) = 0.1013897 eV with x = 0.25). Then, the miscibility gap is analytically estimated and its behaviour as a function of temperature is given by the binodal line, as shown in Fig. 6 (solid line) [24, 25]
The region below the spinodal line is unstable (as indicated by doted line)
kBT / AH0 = 8x(l- x) .
(9)
Fig. 6. T-xphase diagram of Ca1-xMgxO alloys, solid line: binodal curve, doted line: spinodal curve
The critical temperature of the miscibility gap is thus
TGM = 2AH°/kB = 3133 K.
The spinodal curve in the phase diagram marks the equilibrium solubility limit, i.e., the miscibility gap. For temperatures and compositions above this curve a homogeneous alloy is predicted. The wide range between spinodal and binodal curves indicates that the alloy may exist as metastable phase.
Conclusion
In summary, we performed accurate FP-LAPW calculations to investigate the structure and the stability of Ca1-xMgxO solid solutions in the ordered and disordered forms. It is shown that the lattice constants and bulk moduli exhibit a strong deviation from the linear law. The compressibility of the alloys decreases when the composition x increase. The calculated formation energies for both ordered and disordered phases are positive and yield to a miscibility gap. Then, the origin of phase separation was determined to be of a chemical nature. Further, the thermodynamic phase diagram is calculated and a critical temperature of TGM = = 3133 K is found.
International Scientific Journal for Alternative Energy and Ecology № 6 (62) 2008
© Scientific Technical Centre «TATA», 2008
R. Miloua, F. Miloua, Z. Kebbab, N. Benramdane. First-principles investigation the phase separation in Ca1-xMgxO alloys
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