first-principles investigation of the al-si-sr ternary system: ground...

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Intermetallics 21 (2012) 31e44

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Intermetallics

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First-principles investigation of the AleSieSr ternary system: Ground statedetermination and mechanical properties

A.M. Garay-Tapiaa,b,c,*, Aldo H. Romeroa, Gerardo Trapagaa, Raymundo Arróyaveb,c

aCINVESTAV-Unidad Quéretaro, Libramiento Norponiente 2000, CP 76230, Quéretaro, MexicobDepartment of Mechanical Engineering, Texas A&M University, ENPH 119, Mail Stop 3123, College Station, TX 77843-3123, USAcDepartment of Materials Science and Engineering Program, Texas A&M University, ENPH 119, Mail Stop 3123, College Station, TX 77843-3123, USA

a r t i c l e i n f o

Article history:Received 20 June 2011Received in revised form27 August 2011Accepted 4 September 2011Available online 4 November 2011

Keywords:A. Ternary alloy systemsA. Silicides, variousB. Thermodynamic and thermochemicalpropertiesC. Phase diagram, prediction(including CALPHAD)E. Ab-initio calculations

* Corresponding author. Department of MechanicUniversity, ENPH 119, Mail Stop 3123, College Station

E-mail address: [email protected] (A.M. Garay-T

0966-9795/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.intermet.2011.09.001

a b s t r a c t

Twelve different ternary phases of the AleSieSr system were characterized using first-principlescalculations based on Density Functional Theory. Two different exchange correlation functionals wereconsidered to extract its influence on the global results. In addition to the four phases (AlSiSr, Al2Si2Sr,Al2Si2Sr3 and Al16Si30Sr8) reported for this system, we explored eight other ternary phases (Al2Si3Sr3,Al2Si4Sr3, Al2Si7Sr5, Al3Si7Sr10, Al6Si3Sr20, Al6Si9Sr10, Al8Si3Sr14 and AlSi6Sr4) that has been reported inchemically analogous systems. The mechanical stability of the phases was determined through theanalysis of the elastic constants, and the energetic stability was established using the calculatedformation enthalpy. The thermodynamic and vibrational properties of the AlSiSr, Al2Si2Sr3 and Al2Si4Sr3phases were studied considering the harmonic, quasi-harmonic as well as simple anharmonic correc-tions. All phases were found to be stable, except for the AlSi6Sr4 and Al6Si9Sr10. An isothermal section ofthe phase diagram was calculated at 0 K with the Thermocalc software, where it was found that theAl2Si4Sr3 and Al6Si3Sr20 phases can belong to the ternary ground state. The surface energy sections showthat the other phases are metastable with a DG0 K z 2 kJ/mol�atom, all stables and metastable phasesshow a metallic behavior and high compressibility.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The AleSieSr system is very important in the light-metal alloysindustry. In AleSi cast alloys, for example, strontium is used asa modifier of the AleSi eutectic [1]. When small quantities ofstrontium (hundreds of ppm at most) are added to AleSi eutecticalloys, the morphology of the silicon crystallites change dramati-cally, from an acicular to a fibrous morphology. This morphologicalchange refines the microstructure, which in turn results in muchbetter mechanical and structural properties [2]. Furthermore,recent work on AleMg alloys that use silicon as an alloyingelement, shows that strontium improves fatigue resistance at hightemperatures [3e5]. Therefore, a deeper understanding of theAleSieSr ternary system is important for the further developmentof these classes of light metal alloys, which have received consid-erable attention in recent years.

An in-depth study of the phase stability of the AleSieSr ternaryis further justified by considering that several intermetallic

al Engineering, Texas A&M, TX 77843-3123, USA.apia).

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compounds in this system are being studied for different applica-tions: AlSiSr for its superconducting behavior [6,7], the non-stoichiometric SrAl2 [8] for its potential as a hydrogen storagematerial and the Al2Si2Sr [9] and Al16Si30Sr8 [10] phases for theirthermoelectric properties. All of the previously mentioned phaseshave yet to be studied completely and outside of these composi-tions there is little theoretical or experimental informationavailable.

Investigations on the phase stability of chemically-relatedsystems suggest that it may be possible to find new intermetalliccompounds in regions of the AleSieSr phase diagram that are yetto be investigated in detail. Specifically, studies on the AleGeeBa,AleGeeCa, AleGeeSr, and AleSieMg systems report the existenceof several ternary intermetallic phases that could also be stable inAleSieSr. Table 1 shows the crystallographic information of eightintermetallics structures observed in the four systems alreadymentioned along with the structures reported for the AleSieSrsystem. The structures with space group P�6m25p6/mmm,P�3m1 and Immm are present in AleSieCa, AleGeeBa, AleGeeSrand AleSieSr ternary systems [6,11e14] and have similar proper-ties due to the affinity between the elements of the group IV andthe rare-earth elements. The fact that these three structures are

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Table 1Crystallographic information of the intermetallics phases in systems chemicallyanalogous to AleSieSr.

Phase Space group PearsonSymbol

Cell Parameters

Number Symbol a b c

AlSi6Mg4 [15] 12 C2/m mC11 16.53 3.63 6.72Al2Ge3Ca3 [16] 62 Pnma oP8 11.39 4.34 14.83Al6Ge9Ca10 [17] 166 R-3m hR35 13.98 13.98 21.07Al6Ge13Ca20 [17] 176 P63/m hP39 16.00 4.58 16.00Al2Ge4Sr3 [17] 12 C2/m mC9 12.67 4.16 8.87Al8Ge3Sr14 [18] 148 R-3 hR35 11.96 11.96 40.10Al2Ge7Ba5 [12] 12 C2/m mC14 8.59 10.31 18.47Al3Ge7Ba10 [19] 193 P63/mmm hP20 9.74 9.74 16.47Al16Si30Sr8 [14] 223 Pm-3n cP54 10.47 10.47 10.47AlSiSr [14] 187 P-6m2 hP3 4.23 4.23 4.74Al2Si2Sr [14] 164 P-3m1 hP5 4.18 4.18 7.41Al2Si2Sr3 [14] 71 Immm oI7 4.07 4.82 18.99

A.M. Garay-Tapia et al. / Intermetallics 21 (2012) 31e4432

common to all these ternary systems suggests that structures withthe same symmetry and stoichiometry may also be present in theAleSieSr system.

The aim of this study is to investigate the possible existence ofnewphases in the AleSieSr ternary system. In order to achieve this,we investigate the stability of those phases, considering bothenergetic and mechanical criteria. The energetic stability is evalu-ated considering the enthalpy of formation of the intermetallicscalculated from first-principles at 0 K. In this case, there are noentropic contributions and the free energy simply corresponds tothe enthalpy. This methodology was used in the case of the binaryintermetallics of the AleSr [20] and SreSi [21] systems. In thosecases as well as in this work, first-principles calculations werecombined with the CALPHAD approach [22] to obtain the groundstate and corresponding low-temperature phase diagram of thesystem. The phases under consideration include the known inter-metallics in the AleSieSr system, as well as those with the samesymmetry observed in AleXeY (X¼ Si,Ge Y¼Mg,Ca,Sr,Ba) systems(see Table 1). We report on the total energy calculations as well ason the formation enthalpy of these structures. The stability of thesestructures were further investigated by considering mechanicalstability conditions in terms of the elastic tensor [23]. This analysisallowed us to determine whether any of these phases wereunstable and thus not likely to form even under non-equilibrium(metastable) conditions. Finally, for two intermetallic compoundsthat belong to the ternary ground state, we calculated the vibra-tional and thermodynamic properties at finite temperatures.

The paper is organized as follows: Section 2 contains a descrip-tion of the methodology used as well as the details of the first-principles calculations performed. The results and discussion ofthe stability and possible existence of new phases is presented inSection 3, as well as a detailed theoretical description of the stableintermetallics. The discussion of the vibrational properties for theAlSiSr, Al2Si2Sr3 and Al2Si4Sr3 intermetallics is presented in Section4, and the conclusions of the paper are included in Section 5.

Table 2Chosen K-points mesh for the intermetallics studied in this paper, according toa Gamma-centered (G) Monkhorst-Pack.

Phase Mesh (k) Phase Mesh (k)

AlSiSr 16 � 16 � 14 Al2Si7Sr5 8 � 8 � 8Al2Si2Sr 15 � 15 � 9 Al3Si7Sr10 8 � 8 � 4Al2SiSr3 12 � 12 � 12 Al6Si13Sr20 4 � 5 � 15Al16Si30Sr8 4 � 4 � 4 Al6Si9Sr10 6 � 6 � 6Al2Si3Sr3 6 � 15 � 4 Al8Si13Sr14 6 � 6 � 6Al2Si4Sr3 12 � 12 � 9 AlSi6Sr4 9 � 9 � 12

2. Methodology

2.1. Total energy calculations

The total energy calculations were performed within theframework of the Density Functional Theory [24], as implementedin the Vienna Ab Initio Simulation Package (VASP) [25,26]. Twoapproximations for the exchange correlation were considered: theLocal Density Approximation (LDA) according to the Per-deweZunger parameterization [27] and the Generalized GradientApproximation (GGA) in the form of PW91 [28]. Wave functions

were expanded in plane waves, using the Al:3s23p1, Si:3s23p2 andSr:4s24p65s2 electronic configuration for the valence electronsdescribed within the projector augmented-wave (PAW) pseudo-potentials formalism [29], implemented in the VASP package. TheBrillouin zone integrations were performed using a MonkhorstPackmesh [30]. Full relaxations were realized by using the Methfes-selePaxton smearing method of order one [31] and a final self-consistent static calculation with the tetrahedron smearingmethod with Blöchl corrections [32]. To ensure the convergence inenergy to better than 1 meV and 1 kBar in pressure, we used anenergy cut-off of 350 eV and a k-point grid sets according to Table 2.

2.2. Elastic constants calculations

The elastic constants of all the intermetallic compoundsconsidered in this work were calculated using the efficient strain-stress method [33], where a set of strains 3¼ ( 31, 32, 33, 34, 35, 36) isimposed on a crystal structure. If A is the lattice vectors specified inCartesian coordinates, 31, 32, 33 and 34, 35, 36 are the normal strains andthe shear strains respectively. Then, the deformed lattice vectorsare:

A ¼ A

0BBBBB@

1þ 3136

235

236

21þ 32

34

235

234

21þ 33

1CCCCCA

(1)

After the application of the strain, it is possible to obtain the setof stresses t, which result from the changes on the energy due to thedeformation, which is obtained from the electronic structurecalculations. Applying Hooke’s Law (t ¼ 3C), if a n set of strains( 31/n) are applied, the result is n set of stresses (t1/n) which can beused to find the 6 � 6 elastic constant matrix (C) according to:

0@C11 . C16

« 1 «C61 . C66

1A ¼

0@ 31;1 . 31;n

« 1 «36;n . 36;n

1A

�10@ t1;1 . t1;n

« 1 «t6;1 . t6;n

1A (2)

The elements in the C tensor are written as Cij according to theVoigt’s notation and the linearly independent sets of strainsnecessary to calculate the elastic constants can be written of thefollowing way:

3¼

0BBBBBB@

x11 0 0 0 0 00 x22 0 0 0 00 0 x33 0 0 00 0 0 x44 0 00 0 0 0 x55 00 0 0 0 0 x66

1CCCCCCA

(3)

According to the symmetry of the structure, the minimumnumber of linearly independent sets of strains that can be used todetermine the elastic constants are two for cubic, three for

A.M. Garay-Tapia et al. / Intermetallics 21 (2012) 31e44 33

hexagonal and rhombohedral, four for tetragonal, and six fororthorhombic, monoclinic, and triclinic structures.

2.3. Thermodynamic properties

Most thermodynamic properties at finite temperatures can bederived from the structural total free energy. In the present work,we consider the contributions to the total free energy coming fromthe electronic (Fele) and the vibrational free energy (Fvib). Theelectronic part can be calculated according to the one-electronband approximation. This approximation is based on the self-consistent integration of the electronic density of states [34]:

Fele ¼ Eele � TSele (4)

EeleðV ; TÞ ¼Z

hð 3;VÞf 3v 3�Z 3f

hð 3;VÞ 3v 3 (5)

SeleðV ; TÞ ¼ �kB

Zhð 3;VÞ½f �lnf þ ð1� f Þlnð1� f Þvh (6)

h( 3,V) is the electronic density of states and f is the Fermi distri-bution function. 3f is calculated self-consistently to ensure that ateach temperature the total number of electrons is conserved.

The vibrational contributions to the free energy of the crystal(Fvib), as a function of temperature is calculated according to:

FvibðTÞ ¼ kBTZN

0

ln�2sinh

�hn

2kBT

��gðnÞvn (7)

where g(n) is the phonon density of states. This function is usuallyobtained by two main different approaches: density functionalperturbation theory [35] and the (direct) supercell approach [36].In this work, we have use the latter method as implemented in theATAT software package [37]. In this method, the force constants areobtained by calculating the interatomic forceseusing the electronicstructure codedresulting from finite atomic displacements fromthe equilibrium position. The force constants are used to obtain thedynamical matrix which is then diagonalized. In the harmonicapproximation, the resulting eigenfrequencies correspond to thenormal (independent) modes of oscillation of the crystal, while theeigenvectors correspond to the atomic displacements of the indi-vidual modes. The supercell method excludes the anharmoniceffects due to phononephonon or electronephonon interactions orhigher other effects. In the present paper we evaluated the anhar-monic effects originating from phononephonon interactions usingthe expression developed by Wallace and Callen [38]:

Fanh ¼ A2T2 þ A0 þ A�2T þ. (8)

Currently, we only are able to evaluate the first term of theexpansion A2 using information that can be derived directly fromfirst-principles:

A2 ¼ 3kBQ

½0:0078ðgÞ � 0:0154� (9)

where Q corresponds to the harmonic Debye temperature and g

corresponds to the Gruneisen parameter. We would like to pointout that Eq. (9) was obtained from correlations between the Debyeand Gruneisen values for crystals of different symmetries obtainedby Wallace and Callen [38]. Eq. (9) is classical and therefore itcannot be applied at low temperatures. In order to obtain a validexpression for the anharmonic free energy over the entiretemperature range, is necessary a better description of anharmonic

free energy. Using thermodynamic perturbation theory [39] andthe time-averaged internal energy at constant T given by Einstein’sformula:

hEi ¼ 12kBqþ

kBqexpðq=TÞ � 1

(10)

is possible to evaluate the anharmonic free energy, consideringa weakly anharmonic oscillator, symmetric potential and neglect-ing higher terms than proportional to T2 according to (for moredetails see Ref [40]):

Fanh ¼ hU � U0i0 ¼ AT2 ¼ 3nA2

6kB

hhEi2þ2kBCVT2

i(11)

Fanh ¼ hkB$A2

4$

0BB@12Qþ Q

exp�Q

T

�� 1

1CCA

2

þ 2hkB$A2

4$

�Q

2

�2 exp�Q

T

��exp

�Q

T

�� 1

�2T2 ð12Þ

Here, Q corresponds to the high-temperature harmonic Debyetemperature, defined as Q ¼ ðZ=kBÞ½ð5=3ÞhW2i�1=2, Eq. (12) iscalculated at each volume considered in the quasi-harmoniccorrection, with the parameter A2 calculated according to Eq. (9),with q and g calculated directly from first-principles. The thermo-dynamic properties at finite temperatures can be calculated fromthe free energy now defined as:

FðV ; TÞ ¼ E0 KðVÞ þ FvibðV ; TÞ þ FeleðV ; TÞ þ FanhðV ; TÞ (13)

3. Stability of the AlxSiySr1LxLy phases

3.1. Formation enthalpy

In order to discriminate the stability of the different phases inthe AleSieSr system, we first calculate their formation enthalpy.For the four ternary intermetallic phases reported in Ref. [14], thecrystallographic information used as input for the calculations istaken from previous reports (see Table 1). According to Table 3, theexperimental and calculated cell parameters differs by less than 2%.Our calculated formation enthalpy for these phases is shown in thesame table. We would like to note that in this case, the approxi-mation chosen for the calculation of the formation enthalpies doesnot influence the results in a significant manner. In fact, accordingto investigations by the authors on the SreSi system [21,41], we cansuppose that the ‘true’ formation enthalpy of these phases liesbetween the LDA and GGA approximations, as the former usuallyoverestimates the cohesive energies while the latter one tends tooverestimating them.

The formation enthalpy was calculated according to thefollowing equation:

DHAlXSiYSr1�X�Yf ¼ EAlXSiySr1�X�Y � X$EFccAl � Y,EDia�A4

Si

� ð1� X � YÞ$EFccSr (14)

where EAlXSiySr1�X�Y , EFccAl EDia�A4Si and EFccSr are the energies of the pure

elements in the more stable phase at 0 K and EAlxSiySr1�x�y corre-sponds to the total energy of the intermetallic compound at 0 K.The energies of the structures are normalized by the number ofatoms in their respective unit cell.

Table 3Calculated cell parameters and formation enthalpy (kJ/mol-atom) of experimentally observed AleSieSr intermetallics using LDA and GGA approximations.

Phase DHLDAf DHGGA

f Cell parameters

LDA GGA Exp

a b c a b c a b c

AlSiSr �40.85 �43.43 4.21 4.21 4.57 4.26 4.26 4.73 4.23 4.23 4.74Al2Si2Sr �31.00 �31.38 4.12 4.12 7.25 4.20 4.20 7.43 4.18 4.18 7.41Al2Si2Sr3 �42.05 �43.34 4.13 4.69 18.90 4.20 4.83 19.23 4.07 4.82 19.00Al16Si30Sr8 �16.00 �16.20 10.46 10.45 10.44 10.63 10.62 10.62 10.47 10.47 10.47

Reference states: Fcc for Sr and Al, diamond cubic for Si.


Table 3 shows that the formation enthalpy is negative for thefour phases, demonstrating that these phases are energeticallystable. In the case of the AlSiSr and Al2Si2Sr3 phases, we noticea very negative formation enthalpy, which means that at lowtemperatures these phases are very likely to be always present inthe phase diagram andmust therefore belong to the ground state of

Fig. 1. Crystallographic information and unit cel

the AleSieSr system. In the present paper, we propose eight newphases that can belong to the AleSieSr system, based on infor-mation from chemically analogous systems. Fig. 1 summarizes thecrystallographic information of the intermetallics we have alsoincluded in this study. The initial symmetries and cell parameters ofthese phases in the AleSieSr system were chosen according to

l of the considered AlxSiySr1�X�Y structures.


prototype structures in those chemically-analogous systems (seeTable 1).

The AleSieSr ternary phases present interesting properties dueto their crystalline structures. The main structural feature is thatthey contain a skeleton composed of AleSi bonds, which is sur-rounded by strontium atoms. In this AleSi skeleton the silicon atommay have different coordination number (two or four) while thealuminum atoms presents normal coordination number of three.The AleSi bonds have an average of 2.46 � 8 Å and the SieAleSiaverage angle is 116.5 � 10�. The SreSr distance can vary from3.32 Å to 4.69 Å depending on the symmetry of the crystalstructure.

All the new phases were fully relaxed in order to obtain thelowest energy structure and Table 4 shows the results for the finaloptimized structures. The calculated cell parameters are similar tothe prototype phases (see Table 1), with small differences due tothe variation of the atomic radius between different anions (Si orGe) and different cations (Ca, Ba, Mg and Sr). The symmetry wasconserved after the relaxation and the formation enthalpy isnegative for all the phases except for the Al6Si9Sr10 and AlSi6Sr4phases, which have a slightly positive formation enthalpies. Themagnitude of the formation enthalpies for the six energeticallyfavorable phases is in average larger than those calculated from thefour reported phases and discussed previously, suggesting thatsome of these phases may belong to the ground state of theAleSieSr system.

3.2. Mechanical stability

In order to determine if the new phases can exist at finitetemperatures, it is necessary to establish their mechanical stability.A simple way to determine the limit of mechanical stability, is byensuring that the strain energy (1/2Cij 3i 3j) of a given crystal isalways positive for all possible values of the set { 3i} [38], accordingto:

DEV0

hEf 3ig�E0

V0¼

�1� V

V0

�PðV0Þþ

12

0@X6

i¼1

X6j¼1

Cij 3i 3j

1AþO

�n33i

o�

(15)

Where the crystal energy is a quadratic function of the strain, V0 isthe volume of the unstrained lattice, E0 is the total minimumenergy of the unstrained volume, P(V0) is the pressure of theunstrained lattice and finally V is the volume of the strained lattice.The quadratic form of (1/2Cij 3i 3j) imposes further restrictions on theelastic constants Cij depending on the crystal structure, because itmust be positive for all real values of strain. The mechanical

Table 4Calculated cell parameters and formation enthalpy (kJ/mol-atom) using LDA and GGA ap

Phase DHLDAf DHGGA

f Cell parameters

LDA

a b

Al2Si3Sr3 �40.29 �36.56 11.566 4.3Al2Si4Sr3 �43.18 �38.41 12.082 4.1Al2Si7Sr5 �42.87 �38.02 8.046 9.5Al3Si7Sr10 �42.48 �40.41 9.145 9.1Al6Si13Sr20 �39.91 �37.35 16.39 4.6Al6Si9Sr10 1.90 2.10 14.105 14.1Al8Si3Sr14 �29.01 �29.13 11.67 11.6AlSi6Sr4 5.55 7.17 19.357 3.9

Reference states: Fcc for Sr and Al, diamond cubic for Si.After relaxation the space group of Al6Si13Sr20 phases is, number: 6 and symbol: Pm.

stability conditions are shown in the Supplementary materialssection.

Table 5 shows the calculated elastic constants for all the crys-talline systems considered in this work. It can be seen that for allthe phases the Ci¼j

ij components of the strain tensor are positive. Forthe monoclinic phases, the values of the Cis5

i5 and the C46 are closeto zero. In the same manner, we appreciate small values of the Cis5

i5and C46 for the Al16Si30Sr8 and Al8Si3Sr14 phases, which means thatthere is a small distortion in their symmetry.

For the hexagonal phases, the stability conditions (see theSupplementary materials) are satisfied when C11, C33 and C44 arepositive and they are significantly larger than C12 and C13. We cansee that for all hexagonal phases the C11, C33 and C44 values arepositive and at least two times larger than C12 or C13. Therefore, allhexagonal phases in the AleSieSr are mechanically stable. Ina similar fashion, for the orthorhombic phases the stability isguaranteed when all Ci¼j

ij values are positive and Ci¼jij are greater

than 2 times Cisjij . Hence, the orthorhombic phases are also stable

as shown in Table 5.The case of the monoclinic phase happens to be more complex,

owing to the fact that there are 13 independent elastic constants.However, we can summarize the stability conditions as follows: ifthe six elastic constants Ci¼j

ij are positive, coupled to the conditionthat CiiCjj > C2

ij mechanical stability is ensured. In particular, ourcalculation indicates that the monoclinic structure follows all thoseconditions, while C12, C13, C15, C23 and C25 have small values and Ci¼j

ijhave large values. Therefore, we expect that Al2Si4Sr3 and Al2Si7Sr5are both mechanically stable. For the AlSi6Sr4 phase we canappreciate that the C23 and C25 values are larger than C44, C55 andC66 values, which indicates that this phase is in the edge of beingmechanical unstable.

For the Al16Si30Sr8 phase, the small differences between the setof {C11,C22,C33}, {C44,C55,C66} and {C12,C13,C23} show that the struc-ture is quite close to being cubic. The stability conditions aresatisfied because {C11,C22,C33}, {C44,C55,C66}, {C12,C13,C23} are posi-tive and {C11,C22,C33} > {C12,C13,C23}.

Our analysis is able to discriminate the large variation in elasticbehavior within the possibility of having most of our consideredstructures as elastically stable.

3.3. Ternary ground state

An important point in the analysis of phase stability is to knowthe ground state of the system since the presence or absence ofa given phase from the ground state is determined by the relativestability of other phases, which in turn depends of the energy asa function of the composition. In the case of systems with morethan two components, determination of the ground state happens

proximations for the new proposed phases in the AleSieSr system.

GGA

c a b c

88 14.956 11.849 4.453 15.36502 8.559 12.484 4.152 8.75390 17.421 8.232 9.831 17.79545 15.351 9.344 9.344 15.74331 16.37 16.77 4.76 16.7005 21.48 14.39 14.39 227 39 11.965 11.965 39.923 6.829 19.984 3.990 7.022

Table 5Elastic constants calculated (in GPa) by LDA and GGA approximations.

Phase C11 C22 C33 C44 C55 C66 C12 C13 C23 C15 C25 C35 C46

AlSiSrLDA 147.2 147.2 79.31 45.63 45.63 46.80 53.65 26.21 26.21 0.00 0.00 0.00 0.00GGA 131.7 131.7 70.29 37.89 37.89 43.39 44.97 22.73 22.73 0.00 0.00 0.00 0.00Al2Si2SrLDA 134.7 134.7 104.2 38.61 38.61 44.33 46.06 33.40 33.40 0.00 0.00 0.00 0.00GGA 110.3 110.7 72.53 11.42 11.42 43.41 23.54 49.14 49.14 0.00 0.00 0.00 0.00Al2Si2Sr3LDA 122.4 98.59 75.75 29.05 40.18 29.21 25.57 24.68 27.23 0.00 0.00 0.00 0.00GGA 110.9 85.64 71.91 26.87 33.95 28.84 21.79 19.15 23.86 0.00 0.00 0.00 0.00Al16Si30Sr8LDA 92.18 93.72 89.39 28.30 26.75 26.93 55.29 58.91 59.51 1.27 �2.9 0.89 �2.4GGA 92.06 91.29 90.96 30.28 28.54 30.15 48.45 47.98 48.88 1.93 0.31 �0.6 �1.4Al2Si3Sr3LDA 90.70 113.5 91.06 59.87 27.47 46.72 30.87 22.71 43.88 0.00 0.00 0.00 0.00GGA 52.14 86.58 49.74 50.57 17.13 41.91 37.41 42.26 36.56 0.00 0.00 0.00 0.00Al2Si4Sr3LDA 84.12 134.5 134.3 40.74 35.79 61.16 56.43 23.83 22.36 3.40 �5.2 �5.1 �4.2GGA 74.74 85.60 131.7 54.68 26.99 44.09 54.38 7.639 32.73 8.46 �10. �8.4 9.44Al2Si7Sr5LDA 100.4 96.15 107.5 35.29 36.93 30.87 28.95 32.49 29.93 �0.8 1.83 �4.6 2.97GGA 87.00 83.69 91.46 30.89 32.94 27.46 24.34 28.82 25.77 �0.8 1.68 �4.1 2.47Al3Si7Sr10LDA 84.60 84.60 78.94 16.32 16.32 20.89 30.81 21.35 21.35 0.00 0.00 0.00 0.00GGA 75.28 75.28 69.25 17.45 17.45 24.18 26.92 17.95 17.95 0.00 0.00 0.00 0.00Al6Si13Sr20LDA 64.32 67.17 66.98 31.15 25.97 19.26 24.91 23.13 24.92 0.00 0.00 0.00 0.00GGA 55.92 58.05 60.56 27.40 24.26 18.31 20.88 19.18 21.88 0.00 0.00 0.00 0.00Al6Si9Sr10LDA 100.3 101.0 86.40 29.78 29.73 33.85 33.73 23.45 23.42 0.00 0.00 0.00 �0.3GGA 88.20 88.20 77.46 25.32 25.32 29.90 28.40 19.31 19.31 0.00 0.00 0.00 0.00Al8Si3Sr14LDA 56.03 56.03 64.64 21.48 21.48 16.57 22.89 21.37 21.37 0.25 �0.2 0.00 �0.2GGA 49.26 49.26 57.90 19.93 19.93 15.02 19.22 18.83 18.83 0.16 �0.1 0.00 �0.1AlSi6Sr4LDA 71.27 96.10 76.46 34.24 8.449 2.180 39.40 26.76 41.92 5.02 �5.0 1.61 0.65GGA 63.16 85.05 71.43 26.06 8.890 0.649 32.81 21.83 30.73 3.74 �2.6 3.19 0.63


to be quite complex. In this particular work, we determined theground state by globally minimizing the free energy of theAleSieSr system at 0 K, considering all the possible monoatomic,binary and ternary phases likely to take part in equilibria. Theminimization was performed using the Thermocalc software [42]and the CALPHAD methodology [22]. According to this method,the Gibbs free energydor Helmholtz free energy as in condensedphases they are almost identicaldfor the stoichiometriccompounds is written as:

Gm � HSERm ¼ aþ b$T þ c$T$lnðTÞ þ

Xdi$T

i (16)

where the Gm � HSERm is the Gibbs energy relative to a standard

element reference state (SER), HSERm is the enthalpy of the element

in its stable state at the temperature of 298.15 K and the pressure of1 bar, and a,b,c and di are model parameters fit to thermochemicalproperties. At 0 K only the parameter a is important and it can beobtained from a first-principles calculation. The other importantquantity is HSER

m , which corresponds to the reference state of thesystem. In our case, it was necessary to change the reference stateto 0 K according to:

HSERm ð0 KÞ ¼ HSER

m ð298 KÞ � DHSERm (17)

The HSERm ð298 KÞ is taken from the SGTE [43,44] database and

the DHSERm can be obtained following the same line as in Ref [41].

The binary Gibbs energies were taken from reference [20] for theAleSi system and [21] for the SreSi system.

Fig. 2 shows the results of the calculated isothermal sections inboth approximations. Only tree of the four phases reported belong

to the ternary ground state as expected, because the formationenthalpy of the Al16Si30Sr8 is much less negative than other phases.

The main difference between the LDA and GGA calculations isthe fourth phase that belongs to the ternary ground state. In thecase of the LDA approximation, the Al8Si3Sr14 phase belongs to theground state, while the GGA calculations indicate that Al2Si4Sr3belongs to the ground state instead. This difference is due to thephases present in the AleSr binary ground state, which actuallychange depending on whether the LDA or GGA approximations areused [20]. In Fig. 2 it is possible to see that in the LDA approxima-tion the Al7Sr8 phase belongs to the binary ground state while inthe GGA approximation this is not the case.

Fig. 3 shows the (interpolated) energy surface calculated for theternary ground state ( (a) and (c)). According to this figure, themorenegative energy region of the ternary phase diagram is located inthe middle of the SreSi system, corresponding to the SrSi phase.The energy difference between the AleSr and SreSi ground state isaround �16 kJ/mol. This big difference implies that structures withlow amounts of strontium are less stable relative to structures richin Si and Sr. The AleSi side in these diagram corresponds to thepure elements and for this reason the formation enthalpy is zero.The phase closest to the AleSi side is Al2Si2Sr phase, which meansthat phases with less than 20%Sr are not stable at low temperatures.This is the case for the Al30Si16Sr8 phase, which does not belong tothe ground state according to our free energy minimizationcalculations. Fig. 3(b) and (d) show the (interpolated) surfaceenergy calculation for the other phases (those not belonging to theground state), with the energy contour being at least 4 kJ/mol largerthan the energy surface of the ground state. We can appreciate thatfor all these phases the energy difference to the ground state is

Fig. 2. Calculated isothermal sections of the ternary phase diagram at 0 K by (a) LDA and (b) GGA approximations.


between 2 and 4kJ/atom. In other words, the six phases in Fig. 3 (b)and (d) are metastable phases and their existence as equilibriumphases at finite temperatures depends on entropic contributions.

3.4. Electronic properties

In this section, we discuss the electronic properties for theAleSieSr phases considered here, except for the AlSiSr phase,which has been widely studied [45e47]. The Al2Si2Sr phase has

Fig. 3. Calculated energy surface of the ternary phase diagram at 0 K. (a) Ground state surphases those not belonging to the ground state. (c) Ground states surface energy sectionbelonging to the ground state. The numbers are the phases according to: 1-Al2Si2Sr, 2-AlSiSr, 3Al16Si30Sr8.

been recently discussed by some of the present authors [48], theAl16Si30Sr8 where a more detailed study is required and is beyondthe scope of this paper and the AlSi6Sr4 which is unstable asdemonstrated previously in this paper.

We start by presenting the results of the total electronic densityof states, as calculated by using the LDA approximation. Somespecific calculations were performed with the PW-91 exchangecorrelation, but we do not see a significant difference with respectto the LDA approximation, which indicates that it is sufficient to

face energy section by LDA (b) Calculated surface energy section by LDA for the otherby GGA (d) Surface projection of the energy by GGA for the other phases those not-Al2Si2Sr3, 4-Al8Si3Sr14, 5-Al2Si4Sr3, 6-Al6Si13Sr20, 7-Al3Si7Sr10, 8-Al2Si2Sr3, 9-Al2Si7Sr5, 10-

Fig. 4. Partial and total density of state calculated by LDA approximation for different phases in the Al�Si�Sr systems.


Table 6Calculated average charge by Bader analysis.

Phase Al Si Sr % Sr Si%:Al%

Al8Si3Sr14 4.07 6.14 8.92 56.00 3.66:1.00Al6Si13Sr20 0.26 6.92 8.91 51.28 1.00:2.16Al3Si7Sr10 0.00 6.92 8.85 50.00 1.00:3.00Al2Si2Sr3 1.38 7.35 8.84 43.00 1.00:1.00Al6Si9Sr10 0.25 7.11 8.35 40.00 1.00:1.50Al2Si3Sr3 0.00 7.30 8.84 37.50 1.00:1.50Al2Si7Sr5 0.00 5.85 8.84 35.14 1.00:3.50Al2Si4Sr3 0.00 6.38 8.84 33.33 1.00:2.00


investigate the results from only one exchange correlationapproximation.

On the right hand side of Fig. 4, we show the phases withmetallic behavior. For the Al2Si2Sr3 and Al6Si13Sr20 phases, there arep�d cross bands dominating the electronic conduction. For the

Fig. 5. Calculated formation thermodynamic properties of the AlSiSr phase (a) LDA (b) GGAfree energy and Anh: anharmonic contribution.

Al2Si7Sr5 phase, the p and d orbitals are hybridized. However, forthe Al3Si7Sr10 phase, which has more strontium than the previousones, there is a partial hybridization between the p and d orbitalsclose to the Fermi level. At high energies there are more d statesthan p, this is due to the aluminum increase.

On the left hand side of Fig. 4, we report the results for thephases with low density of states at the Fermi level. The Al2Si3Sr3and Al2Si4Sr3 phases have a p�d cross bands and part of the s and porbitals are hybridized. The increase of silicon in the Al2Si4Sr3 phaseshows a decrease in the density of states at the Fermi level withrespect to the Al2Si3Sr3 phase. This is because the increase of siliconcreates more localized electrons between the Al�Si bonds.

The Al6Si9Sr10 and Al8Si3Sr14 phases have a semi-metallicbehavior and in the first one it is also possible to see the p�d crossbands as in the other phases, but in this case only few states areavailable between the valence and conductionbands. The case of theAl8Si3Sr14 phase is totally different, because this structure is

. Har: harmonic approximation, Q-har: Quasi-harmonic approximation, Ele: Electronic


conformed by small pyramids of aluminum inside a matrix ofstrontium and silicon (see Fig. 1). The density of states for thisstructure shows a very small gap at the Fermi level, where electronson the valence band are mostly coming from p orbitals from thealuminum atoms, while the contribution to the conduction band ismostly from d orbitals of the strontium atoms. The above shows thatthe non-localized electron of aluminum cannot go freely betweenaluminum structures and therefore this phase is semi-metallic.

Table 6 shows the charge average calculated through a Bader-type analysis [49,50]. According to this table, in most phases,aluminum is a donor element, but for the Al2Si3Sr3 phase there arefew covalent bonds between silicon and aluminum. In theAl8Si3Sr14 phase the aluminum behavior is different and the tableshows a change of four which means that the aluminum atomshave covalent bonds between them and ionic bonds with thestrontium atoms.

The chemistry of the silicon atoms depends on the concentra-tion of aluminum:when the relation (Al%:Si%) is 1:1� Si%� 1.5, the

Fig. 6. Calculated formation thermodynamic properties of the Al2Si2Sr3 phase (a) LDA (b) GGfree energy and Anh: anharmonic contribution.

SieAl bonds have mostly ionic character and when the siliconconcentration increases, the phase has more SieSi bonds and thesilicon atoms can have sp and sp2 hybridization.

4. Vibrational and thermodynamic properties of the AlSiSr,Al2Si2Sr3 and Al2Si4Sr3 phases

The vibrational properties of the AlSiSr phase has been studiedin the past years [51,52] due to its superconducting behavior. Wenow discuss the formation thermodynamic properties of the AlSiSr,Al2Si2Sr3 and Al2Si4Sr3 phases.

Fig. 5 shows the formation enthalpy, entropy and free energy ofthe AlSiSr phase as a function of temperature. These thermody-namic properties were calculated by successively consideringadditional contributions to the total free energy. The Q � har andHar þ Ele curves show that the electronic contribution are of thesamemagnitude as the quasi-harmonic contribution and these twoaccount for almost 90% of the temperature dependence of the

A. Har: harmonic approximation, Q-har: Quasi-harmonic approximation, Ele: Electronic


thermodynamic properties of this structure.We can also notice thatthe formation entropy of this phase is negative, indicating a stiff-ening of the average interatomic bonds as Al, Si and Sr form AlSiSr.This is also in agreement with the exothermic formation enthalpyof this compound. Note, however, that the formation entropy is notvery significant as it amounts to less than 0.25kB and the formationfree energy only increases by about 2 kJ/mol�atom over a 1000 Ktemperature range.

Fig. 6 shows the formation thermodynamic properties of theAl2Si2Sr3 phase. The figure shows that the behavior of this phase isvery similar to AlSiSr, with the only difference that anharmoniceffects seems to be even less significant in this case. The electronicfree energy has the same strong contribution to the total freeenergy when compared to the harmonic and quasi-harmoniccontributions. However, the formation entropy of this phase ismuch larger than that of AlSiSr and Al2Si4Sr3. This in turn increasesthe de-stabilization of this phase as temperature increases. Thisresult is consistent with the observations on the pseudo-binary

Fig. 7. Calculated formation thermodynamic properties of the Al2Si4Sr3 phase (a) LDA (b) GGfree energy and Anh: anharmonic contribution.

SreAl2Si2Sr phase diagram [53], which have shown that theAl2Si2Sr3 phase has a lower melting point than AlSiSr and Al2Si4Sr3.

Fig. 7 shows the formation thermodynamic properties of theAl2Si4Sr3 phase. Here, the main difference with the two previouscases is the large contribution to the free energy due to electronicdegrees of freedom. The electronic free energy, alongwith the othercontributions increase the formation enthalpy by more than 1 kJ/mol�atom as temperature increases. Given the fact that theformation enthalpy of this phase is rather close to that of AlSiSr andAl2Si2Sr3 and its formation entropy is less than that of Al2Si2Sr3 it islikely that Al2Si4Sr3 may become thermodynamically stable at largetemperatures.

Fig. 8 shows the phonon dispersion for the three phases consid-ered in this section. The phonon dispersion for the AlSiSr phase hasbeen previously studied by other groups [45,46,54,55]. Our resultsare in good qualitative agreement with the reported calculations byHuang et al. [54,55], with the maximum frequency of the acousticmodes being approximately 3 THz and the maximum frequency of

A. Har: harmonic approximation, Q-har: Quasi-harmonic approximation, Ele: Electronic

Fig. 8. Phonon dispersion along high-symmetry directions for (a) AlSiSr (b) Al2Si2Sr3and (c) Al2Si4Sr3.

Table 7Calculated Thermodynamic Properties of AlSiSr, Al2Si2Sr and Al2Si4Sr3.

Phase a (10�6 1/K) B (GPa) B0 Gru 298 K Cv 298 K (J/mol/K)

AlSiSrLDA 13.48 52.51 4.189 1.304 23.00GGA 18.17 45.31 4.238 1.576 23.45Al2Si2SrLDA 13.20 45.66 4.234 1.236 23.50GGA 14.97 40.02 4.443 1.302 23.72Al2Si4Sr3LDA 13.71 54.59 4.360 1.324 23.10GGA 15.69 46.81 4.526 1.376 23.38AlLDA 22.40 76.50 4.801 2.182 23.85GGA 26.79 66.39 4.843 2.349 24.40SiLDA 2.776 94.29 4.357 0.541 19.9GGA 3.553 86.40 4.421 0.634 20.0SrLDA 21.22 13.59 3.737 1.036 25.1GGA 27.18 10.68 4.242 1.168 25.4

Note: a is the coefficient of thermal expansion, B is the bulk modulus, B0 is thepressure derivative of the bulk modulus and Gru is the Gruneissen parameter.


the optical modes being around 13�14 THz in both cases. As in theprevious works [54,55], the acoustic modes are rather flat at thecenter of the Brillouin zone. For the other phases, to our knowledgethephonondispersioncurveshavenot been reportedpreviously. TheAl2Si2Sr3 phase shows a gap between the TA acoustic modes and theoptical modes. Another gap can be appreciated at even higherfrequencies, suggesting that this phase may have lower transportproperties than the other two. The last dispersion curve (Fig. 8 c)shows that the Al2Si4Sr3 phase has high cross banding between theacoustic and optical modes. In all three cases there are no unstablemodes in any region of the Brillouin zone, in agreement with themechanical stability analysis presented above.

The summary of the results can be seen in the Table 7. Thethermal expansion coefficients are lower than those of pure

aluminum and strontium, although they are not as low as thethermal expansion coefficient of pure silicon. In the same way, thebulk modulus is much lower than that of pure aluminum andsilicon, due to the sensitivity of strontium atoms to compression.Furthermore, the low Gruneisen parameter indicates that in all ofthese phases there is a weak dependence of the thermodynamicproperties on temperature. This is corroborated by the formationfree energy calculations for these structures, which increases onlyby 2e3 kJ/mol�atom over a 1000 K temperature range. Finally, thebehavior of the heat capacity is similar to that of a metallic phasedue to the rather large contribution of electrons. In fact, thecalculated heat capacities are very similar to the heat capacity ofaluminum.

5. Summary and conclusions

We have studied, by using first-principles calculations, thestructural, elastic and vibrational stability of a large set ofAlxSIySr1�x�y alloys. It was demonstrated the possibility of theexistence of new phases in the Al�Si�Sr system due to theirenergetic and mechanical stability, these results can be summa-rized as follows:

� In this work, the stability of twelve intermetallic compounds inthe Al�Si�Sr system were studied. Four of these intermetallicshave already been reported in the literature and eight of thesestructures have been reported in other chemically-analogoussystems.

� The four reported phases were mechanically and energeticallystable, while of the new phases only six were energeticallystable. Virtually all the compounds studied were shown to bemechanically stable, with the only exception being the AlSi6Sr4phase.

� The Al2Si3Sr3, Al2Si4Sr3, Al2Si7Sr5, Al3Si7Sr10, Al6Si13Sr20 andAl8Si3Sr14 intermetallics have an exothermic formationenthalpy. However, the stability of the phases is determined bythe equilibrium conditions among all the phases in the ternaryAleSieSr system.

� The calculated ground state shows two new possible phases forthe AleSieSr ternary system with the other candidate phasesbeing at best metastable with a DHa/b ¼ 2�4 kJ/mol�atom.

� All the ternary intermetallics in the AleSieSr system havemetallic behavior with only the Al8Si3Sr14 intermetallic having


a semi-metallic behavior. The Al2Si4Sr3 and Al2Si3Sr3 phases aremetallic but with few states in the Fermi level.

� The vibrational and thermodynamic properties of the AlSiSr,Al2Si2Sr3 and Al2Si4Sr3 were studied considering the quasi-harmonic, anharmonic and electronic contribution to the freeenergy within the LDA and GGA approximations. These threephases show high electronic contribution to the free energy,this contribution are strong as the quasi-harmonic contribu-tion. For the three phases we cannot appreciate significantchanges when we employ the anharmonic contribution.

� The bulk modulus and the thermal expansion coefficients ofthe AlSiSr, Al2Si2Sr3 and Al2Si4Sr3 phases are low compared tothe pure elements, while the heat capacity of the three issimilar to the pure aluminum.

Acknowledgments

The authors would like to acknowledge the TAMU-CONACyTprogram for its support. A. H. Romero would like to acknowledgethe support of CONACYT under projects J-152153-F, J-83247-F,TAMU-CONACYT and FNRS-CONACYT Binational collaboration. A.Garay. acknowledges CONACYT (México) for the support throughscholarship number 216341. R. Arróyave acknowledges the supportfrom NSF through grants CMMI-0758298, 1027689, 0953984,0900187 and DMR-0805293. First-principles calculations werecarried out in the Chemical Engineering Cluster as well as the TexasA&M Supercomputng Facility as well as in the Ranger Cluster at theTexas Advanced Computing Center. Preparation of the input filesand analysis of the data have been performed within the frame-work AFLOW/ACONVASP [56] developed by Stefano Curtarolo aswell as with the ATAT package [37,57] developed by Axel van deWalle.

Appendix. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.intermet.2011.09.001.

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