PHYSICAL REVIEW B 79, 134203 共2009兲
First-principles solubilities of alkali and alkaline-earth metals in Mg-B alloys
Roman V. Chepulskii and Stefano Curtarolo*
Department of Mechanical Engineering and Materials Science, and Center for Theoretical and Mathematical Sciences, Duke University,
Durham, North Carolina 27708, USA
共Received 17 February 2009; published 3 April 2009兲
In this article, we present a comprehensive theoretical study of solubilities of alkali 共Li, Na, K, Rb, Cs兲 and
alkaline-earth 共Be, Ca, Sr, Ba兲 metals in the boron-rich Mg-B system. The study is based on first-principles
calculations of solutes formation energies in MgB2, MgB4, MgB7 alloys and subsequent statisticalthermodynamical evaluation of solubilities. The advantage of the approach consists in considering all the
known phase boundaries in the ternary phase diagram. Substitutional Na, Ca, and Li demonstrate the largest
solubilities, and Na has the highest 共0.5%–1% in MgB7 at T = 650– 1000 K兲. All the considered interstitials
have negligible solubilities. The solubility of Be in MgB7 cannot be determined because the corresponding
low-solubility formation energy is negative indicating the existence of an unknown ternary ground state. We
have performed a high-throughput search of ground states in binary Mg-B, Mg-A, and B-A systems, and we
construct the ternary phase diagrams of Mg-B-A alloys based on the stable binary phases. Despite its hightemperature observations, we find that Sr9Mg38 is not a low-temperature equilibrium structure. We also determine two possible ground states CaB4 and RbB4, not yet observed experimentally.
DOI: 10.1103/PhysRevB.79.134203
PACS number共s兲: 61.50.Ks, 74.70.Ad, 81.30.Hd
I. INTRODUCTION
The interest in magnesium diboride emerged after the discovery of superconductivity in MgB2 at about Tc = 39 K.1
Attempts to increase Tc by small additions of alkali 关Li,2–6
Na,7,8 Rb,9,10 Cs9,10兴 and alkaline-earth 关Be,11 Ca,4,7,12,13 Ba9兴
metals to MgB2, proved to be unsuccessful. The difficulty
was attributed not only to the inability of such solutes to
decrease Tc but also to their low solubility and precipitation
in secondary phases. Although a claimed superconductivity
at 50 K was reported for the Mg-B-A 共A = Cs, Rb, Ba兲
system,9 attempts to reproduce the results have been so far
unsuccessful 共for A = Cs, Rb兲.10 The problem can be attributed to solutes’ segregation in grain boundaries and to low
solubility in the bulk phase 共much lower than reported in
Ref. 9兲.
For consistent interpretation of experimental observations,
theoretical studies of solubility in Mg-B of alkali metals and
alkaline-earth metal solubilities are therefore necessary. For
instance, in Ref. 14, the semiempirical Miedema approach15
and the Toop’s model16 were used to address the heats of
formation of binary alloys Mg-A, B-A, and Mg-B 共A = Li, Na,
and Ca兲. It was proposed that Ca may form stable compounds while Na and Li could lead to metastable or unstable
ternary phases in MgB2. In Ref. 17, by calculating firstprinciples formation energies of Li and Na impurities in
MgB2, and by neglecting the effects of other ground states in
the ternary phase diagram solubility calculations, it was concluded that Na should have very low solubility, whereas the
solubility of Li should be comparatively higher although
modestly diminished by a segregation into a LiB phase.
The present paper is orthogonal to previous studies. We
develop a comprehensive theoretical framework to determine
the solubilities of alkali metals 共Li, Na, K, Rb, Cs兲 and alkaline earths 共Be, Ca, Sr, Ba兲 in the boron-rich Mg-B system.
The study consists of first-principles calculations of solutes
formation energies in MgB2, MgB4, MgB7 alloys and subse1098-0121/2009/79共13兲/134203共12兲
quent statistical-thermodynamical evaluations of solubilities
with respect to all known equilibrium states of the Mg-B-A
system. The results help outlining future directions in experimental searches.
The paper is organized as following. In Sec. II, we describe the adopted solubility mechanisms in Mg-B. In Sec.
III, we discuss the relevant impurity formation energies in
terms of supercell energy calculations and the appropriate
ground state共s兲. In Sec. IV, the approximation for the free
energy of Mg-B-A solid solution is formulated. In Sec. V, we
present an approach for solubility calculation considering all
the ternary ground states. A simple analytical low-solubility
approximation is devised. Section VI is devoted to the highthroughput ab initio search for ground states in binary Mg-B,
Mg-A, and B-A systems, to the ternary phase diagrams of
Mg-B-A systems, and to the impurity formation energies determined through the phase boundaries of the systems. The
numerical values of solubilities are presented in Sec. VII.
Section VIII summarizes the results, draws conclusions, and
comments on strategies for future research in this area.
II. Mg-B-A SOLID SOLUTION
For the description of solubility of alkali and alkalineearth metal elements A 共A = Li, Be, Na, Mg, K, Ca, Rb, Sr,
Cs, Ba兲 in the Mg-B system, we consider the disordered
solid solutions of A atoms as interstitial and substitutional
impurities in the experimentally reported compounds18,19
MgB2, MgB4, and MgB7. We consider only substitutions in
the magnesium sublattice as boron substitutions have not
been observed experimentally.2–13 The disordered solid solution of A inside Mg-B is labeled as “共1兲” throughout the
paper.
While the magnesium-substitutional positions are
determined,19 the “most accommodating” interstitial locations have to be found. The task is implemented with the
134203-1
©2009 The American Physical Society
PHYSICAL REVIEW B 79, 134203 共2009兲
ROMAN V. CHEPULSKII AND STEFANO CURTAROLO
TABLE I. Centers and sizes of the symmetrically inequivalent
interstitial cages for the MgB2, MgB4, and MgB7 unit cells. Coordinates are presented as fractions of the standard unit cell basis
vectors 共a , b , c兲 共Ref. 19兲. The radius is the distance from the cage
center to the nearest Mg or B atom. ␯i indicates the numbers of the
symmetrically equivalent cages within the unit cell. The total numbers of distinct geometrical cages before agglomeration upon insertion of an interstitial is reported in brackets.
Structure
MgB2
MgB4
共Mg4B16兲
MgB7
共Mg8B56兲
i
Coordinates
共fraction of a, b, c兲
Cageradius
Å
␯i
1
2
3
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
9
10
0.341, 0.681, 0.000
0.006, 0.000, 0.500
0.500, 0.500, 0.122
0.106, 0.669, 0.412
0.375, 0.250, 0.337
0.145, 0.632, 0.813
0.144, 0.750, 0.925
0.225, 0.750, 0.719
0.086, 0.015, 0.527
0.101, 0.543, 0.261
0.000, 0.049, 0.618
0.189, 0.250 0.821
0.250, 0.217, 0.750
0.243, 0.250, 0.505
0.197, 0.250, 0.469
0.000, 0.250, 0.150
0.012, 0.000, 0.000
0.250, 0.062, 0.250
0.000, 0.171, 0.788
0.131, 0.186, 0.119
1.7619
1.7616
1.6015
1.9145
1.7836
1.7821
1.7252
1.6322
1.6128
1.5322
2.0050
1.8937
1.7995
1.7888
1.7802
1.7002
1.6995
1.6068
1.4991
1.4443
2 共6兲
1 共6兲
6 共6兲
4 共52兲
4 共4兲
8 共24兲
4 共24兲
4 共4兲
8 共8兲
8 共8兲
8 共152兲
8 共24兲
8 共8兲
8 共24兲
8 共24兲
4 共24兲
8 共64兲
8 共40兲
8 共8兲
16共16兲
following exhaustive search performed through our software
AFLOW.20,21 Let us consider a quadruplet of no-coplanar atoms, where the first atom belongs to the unit cell and the
others are closer than the maximum diagonal of the unit cell
to the first atom. A cage is defined when the spherical region
of space touching all the four atoms of the quadruplet does
not contain further atoms inside. An interstitial position is
found if the cage has its center inside the unit cell. By considering all the possible combinations, the symmetrically inequivalent interstitials can be identified through the calculation of their site symmetry 共with the factor group of the unit
cell兲. Note that in unit cells with complex arrangements,
many of the interstitials positions can be extremely close.
Thus, an interstitial atom located in any of those close positions would deform the nearby local atomic environment and
relax to the same final location. Hence, the number of symmetrically inequivalent cages can be further reduced by considering the whole set of positions that would agglomerate
upon insertion of an interstitial atom, as a single interstitial
site. The results of the search are presented in Table I. The
table demonstrates that the higher boron contents the larger
number of cages and the larger radius of the bigger cage.
III. FORMATION ENERGIES DEFINITIONS
A. “Raw” formation energies
Let us define the so-called raw formation energies ␮i and
¯ s 共composition unpreserving22兲 as the changes in the ener␮
gies of the solvent upon introduction of one solute atom A in
the ith type interstitial or sth type substitutional positions. In
first-principles calculations, the solvent can be replaced by a
large supercell 共“sc”兲, so that
共i兲
␮i ⬅ ␮共1兲
i 共A兲 = Esc关A MgnscBmsc兴 − Esc关MgnscBmsc兴,
¯ s ⬅ ␮s共1兲共A兲 − ␮共1兲共Mg兲
␮
= Esc关A共s兲Mgnsc−1Bmsc兴 − Esc关MgnscBmsc兴,
where nsc and msc are the numbers of Mg and B atoms in the
共1兲
supercell; ␮共1兲共Mg兲, ␮共1兲
i 共A兲, and ␮s 共A兲 are the zerotemperature chemical potentials of the Mg atom, and the A
atoms in the ith interstitial and sth substitutional positions,
respectively.23
The conventional unit cells of MgB2, MgB4, and MgB7
are used to construct the appropriate supercells. The supercells are chosen to have solute-solute A-A distance at least
⬃2.5 times the nearest-neighbor solvent bonds in order to
diminish the contribution of the A-A interactions to the calculated energies. We use 2 ⫻ 2 ⫻ 2 unit cells for MgB2,
while there is no need to create supercells of MgB4 and
MgB7 since their unit cells are already large enough. Note
that boron-boron 共B-B兲 bond is the shortest among the MgMg, B-B, and Mg-B bonds. Table II summarizes the parameters of the supercells and Table III lists the supercell atomic
compositions.
B. “True” formation energies
The true formation energies 共composition preserving22兲
are defined from the raw formation energies 共composition
unpreserving22兲 as
TABLE II. Numbers of atoms n , m 共MgnBm兲 in the conventional unit cells 共“uc”兲 共Ref. 19兲 and in the
constructed supercells 共“sc”兲 for MgB2, MgB4, and MgB7. ␯s indicates the number of symmetrically equivalent s-type Mg atom positions within the corresponding unit cell: 兺s␯s = nuc. dAA and dBB are the distances
between the nearest solutes and nearest solvent atoms 共boron-boron in our case兲 in the supercell.
Compound
MgB2
MgB4
MgB7
共1兲
nuc
muc
␯s
nsc
msc
dBB 共Å兲
dAA / dBB
1
4
8
2
16
56
1
4
4,4
8
4
8
16
16
56
1.7811
1.7004
1.7372
3.46
2.60
3.44
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PHYSICAL REVIEW B 79, 134203 共2009兲
FIRST-PRINCIPLES SOLUBILITIES OF ALKALI AND…
TABLE III. Interstitial and substitutional supercell atomic comsc sc
positions 共xsc
A , xB , xMg兲 as functions of the nsc 共for Mg兲 and msc 共for
B兲 numbers of atoms in the super cell.
xsc
A
xsc
B
sc
xMg
Interstitials
Substitutional
关1 + nsc + msc兴−1
msc关1 + nsc + msc兴−1
nsc关1 + nsc + msc兴−1
关nsc + msc兴−1
msc关nsc + msc兴−1
nsc关nsc + msc兴−1
EA共i兲 ⬅ ␮i − ␮Apure ,
pure
¯ s + ␮Mg
EA共s兲 ⬅ ␮
− ␮Apure ,
共2兲
pure
are the chemical potentials 共energies
where ␮Apure and ␮Mg
per atom兲 of A and Mg atoms in pure A and Mg solids at zero
temperature, respectively.
MgB2 with a substitutional Li atom兲 as shown in Fig. 2. The
coefficient Xk 共k = 1 , 2 , 3兲 represents the fraction of the kth
␣
defined in Eq. 共4兲
phase in the mixture. The quantities ⌬Esc/at
are the supercell formation energies 共per atom兲 determined
␣
共A兲 and
with respect to the mixture of ground states.24 Esol
␣
⌬Esc/at cannot be negative. If they do, the list of ground states
is incomplete, and a better phase diagram should be established 共the missed ground state can be the supercell itself兲.24
In analogy with binary alloys,24 the ternary alloy low␣
共A兲 is shown to determine the
solubility formation energy Esol
solubility of A in the low-solubility limit 共Sec. V兲.
IV. FREE ENERGY OF THE Mg-B-A SOLID SOLUTION
The Gibbs free energy per unit cell 共“uc”兲 of the Mg-B-A
solid solution is determined within the mean-field approximation,
共1兲
关兵ci,cs其,T兴 = Euc关MgnucBmuc兴 + ⌬gI + ⌬gS ,
Guc
C. Low-solubility formation energies
⌬gI = 兺 i ␯i兵ci␮i + kBT关ci ln ci + 共1 − ci兲ln共1 − ci兲兴其,
In Ref. 24, a quantity called “low-solubility formation energy,” was shown to determine the dilute solubility in binary
alloys through the temperature exponential factor. The quantity can be generalized to the case of Mg-B-A alloy as 共compare with Eq. 共29兲 in Ref. 24兲,
␣
␣
共A兲 ⬅ ⌬Esc/at
/xAsc ,
Esol
共s兲
=
⌬Esc/at
Esc关A共i兲MgnscBmsc兴
1 + nsc + msc
sc sc sc
− Emix
at 共xA ,xB ,xMg兲,
Esc关A共s兲Mgnsc−1Bmsc兴
nsc + msc
sc sc sc
− Emix
at 共xA ,xB ,xMg兲,
共4兲
sc sc sc
where Emix
at 共xA , xB , xMg兲 is the energy of a the three-phase
mixture at the same composition of the supercell 共see Table
III兲,
sc sc sc
Emix
at 共xA ,xB ,xMg兲 = X1
Euc关MgnucBmuc兴
nuc + muc
共3兲
+ X2E共2兲
at + X3Eat .
共5兲
The coefficients Xk 共k = 1 , 2 , 3兲 are determined from the following linear system of equations,
冦
sc
xA
= X2xA共2兲 + X3xA共3兲
xBsc = X1
muc
+ X2xB共2兲 + X3xB共3兲
nuc + muc
sc
xMg
= X1
muc
共2兲
共3兲
+ X2xMg
+ X3xMg
,
nuc + muc
共7兲
共3兲
with ␣ = 共i兲 , 共s兲 and
共i兲
⌬Esc/at
=
¯ s + kBT关cs ln cs + 共1 − cs兲ln共1 − cs兲兴其,
⌬gS = 兺 s ␯s兵cs␮
冧
where kB is a Boltzmann constant, T is the temperature, and
Euc关MgnucBmuc兴 represents the energy 共enthalpy兲 of initial
MgnucBmuc unit cell without A solutes; the summations over i
and s are with respect to all the inequivalent types of interstitial and substitutional positions in unit cell; ci and cs are
equal to the site concentrations of A atoms at each interstitial
共i-兲 and substitutional 共s-兲 type, respectively. Thus, we assume that the concentrations in the equivalent positions are
equal in the disordered state. The raw formation energies ␮i
¯ s are introduced in Sec. III. The mean-field approximaand ␮
tion, although neglecting correlations, should work well
when the deviation from stoichiometry is small 共see Sec. 19
in Ref. 25兲. In addition, we neglect solute-solute interactions
that might be important especially at high solute concentrations. In conclusion, our model is similar to Wagner-Schottky
model of a system of noninteracting particles.26
For given concentrations 兵ci , cs其, the total number of atat
, the total concentrations of A interstioms per unit cell, Nuc
共1兲
tials, xA共i兲, A substitutional, xA共1兲共s兲, and the total concentration
of A atoms, xA共1兲, are determined as
at
Nuc
= 兺 i ␯ici + nuc + muc ,
共6兲
at
at
,xA共s兲 = 兺 s ␯scs/Nuc
,
xA共i兲 = 兺 i ␯ici/Nuc
共1兲
共1兲
共k兲 共k兲 共k兲
where E共k兲
at , xA , xB , xMg 共k = 2 , 3兲 are the energies 共per atom兲
and stoichiometric compositions of the two other ground
states that, together with MgnucBmuc, form the convex-hull
sc
兲 in the ternary
triangle containing the point 共xAsc , xBsc , xMg
phase diagram. For example, MgB2, Mg, and LiB3 are the
three ground states surrounding Mg7B16Li 共a supercell of
共1兲
共1兲
xA共1兲 = xA共i兲 + xA共s兲 .
共8兲
共1兲
At given temperature and concentration xA
, the Gibbs
共1兲 共1兲
free energy per atom Gat 关xA , T兴 is determined by minimizing Eq. 共7兲 with respect to 兵ci其 and 兵cs其,
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PHYSICAL REVIEW B 79, 134203 共2009兲
ROMAN V. CHEPULSKII AND STEFANO CURTAROLO
TABLE IV. The formation energies ⌬Ebin
at 关Eq. 共27兲兲兴 for Mg-B, B-A, and Mg-A binary ground states
共A = Li, Be, Na, Mg, K, Ca, Rb, Sr, Cs, Ba兲. The symbol 共+兲 indicates possible ground states never observed
experimentally 共CaB4 and RbB4兲. The symbol 共−兲 indicates the experimentally observed Sr9Mg38, which was
not confirmed to be a ground states by first-principles calculations.
⌬Ebin
at 共eV/at.兲
Prototypea
Spacegroupb
Pearson
Ref.
Li3B14
LiB3
−0.219
−0.235
Li3B14
LiB3
I4̄2d共122兲
P4 / mbm共127兲
tI160
tP20
39
40
Li8B7
Be3B50
Be1.11B3
−0.216
−0.032
−0.096
Li8B7
Be3B50
Be1.11B3
P6̄m2共187兲
P42 / nnm共134兲
P6 / mmm共191兲
hP15
tP53
hP111
32
41
42
MgBe13
NaB15
Na3B20
MgB7
MgB4
MgB2
−0.009
−0.059
−0.070
−0.138
−0.152
−0.151
NaZn13
NaB15
Na3B20
MgB7
MgB4
AlB2
Fm3̄c共226兲
Imma共74兲
Cmmm共65兲
Imma共74兲
Pnma共62兲
P6 / mmm共191兲
cF112
oI64
oC46
oI64
oP20
hP3
43
44
45
46
47
48
KB6
−0.043
CaB6
Pm3̄m共221兲
cP7
49
CaB6
CaB4共+兲
CaMg2
RbB4共+兲
−0.423
−0.410
−0.128
−0.163
CaB6
UB4
MgZn2
UB4
Pm3̄m共221兲
P4 / mbm共127兲
P63 / mmc共194兲
P4 / mbm共127兲
cP7
tP20
hP12
tP20
50
51
52
SrB6
Sr2Mg17
Sr9Mg38共−兲
−0.464
−0.055
−0.070
CaB6
Th2Ni17
Sr9Mg38
Pm3̄m共221兲
P63 / mmc共194兲
P63 / mmc共194兲
cP7
hP38
hP94
53
54
55
Sr6Mg23
SrMg2
−0.085
−0.114
Th6Mg23
MgZn2
Fm3̄m共225兲
P63 / mmc共194兲
cF116
hP12
56
57
BaB6
−0.421
CaB6
Pm3̄m共221兲
cP7
53
Ba2Mg17
−0.072
Zn17Th2
R3̄mh共166兲
hR57
54
Ba6Mg23
BaMg2
−0.085
−0.097
Th6Mg23
MgZn2
Fm3̄m共225兲
P63 / mmc共194兲
cF116
hP12
56
57
aReference
19.
Reference 38.
b
共1兲
共1兲
G共1兲
at 关xA ,T兴 = min 关Gat 关兵ci,cs其,T兴兴x共1兲 ,
xA共1兲共nuc + muc兲 = 兺 ␯ici共1 − xA共1兲兲 + 兺 ␯scs .
共9兲
A
兵ci,cs其
i
共12兲
s
where
共1兲
at
G共1兲
at 关兵ci,cs其,T兴 = Guc 关兵ci,cs其,T兴/Nuc .
共10兲
The solution 兵ci , cs其 defines the equilibrium distribution of
interstitial and substitutional A solutes in the MgnucBmuc solvent.
In the case of small concentrations of interstitials, the
minimization can be done with the Lagrange multiplier
method, obtaining
冋
ci = 1 + exp
␮i + ␮共1 − xA共1兲兲
k BT
册
−1
,
冋
cs = 1 + exp
␮s + ␮
k BT
册
−1
.
共11兲
where the Lagrange multiplier ␮ is determined from the following equation 关derived from Eq. 共8兲兴,
V. SOLUBILITY
According to Nernst’s theorem, either a single compound
or a phase separation of compounds at correct stoichiometry
can be present at equilibrium at zero temperature. At finite
temperatures, the composition of phases can differ from stoichiometry through solution because of the entropic promotion 共Sec. VII and Ref. 24兲. At a given temperature, the solubility of A atoms in a compound is defined as the maximum
homogeneously achievable concentration of A atoms, without the nucleation of a new phase.
To calculate the solubility,27 we consider the Gibbs free
of the mixture of three phases with a given
energy Gmix
at
gen
兲. The first
general 共“gen”兲 composition xgen ⬅ 共xAgen , xBgen , xMg
phase is the substitutional and/or interstitial solid solution
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PHYSICAL REVIEW B 79, 134203 共2009兲
FIRST-PRINCIPLES SOLUBILITIES OF ALKALI AND…
with atomic concentration xA共1兲 of element A in MgnucBmuc. At
zero temperature, the two other phases and MgnucBmuc form a
triangle containing the point xgen in the ternary phase diagram. At a finite temperature, the free energy Gmix
at is the
generalization of Eq. 共5兲,
共1兲
共1兲
共1兲
共1兲 共1兲
共2兲
Gmix
at 关xA ,T兴 ⯝ X1关xA ,T兴Gat 关xA ,T兴 + X2关xA ,T兴Eat
+ X3关xA共1兲,T兴E共3兲
at ,
共13兲
is given by Eq. 共9兲, and the second and
where
third phases are assumed to be stoichiometric, so that their
free energies can be approximated by their ground-state en共3兲
ergies 共enthalpies兲 E共2兲
at and Eat , respectively. The approximation does not affect much the results, as we are interested
in the small solubility regime of A. Similarly to Eq. 共6兲, the
fractions Xk are determined by solving the system,
冦
xAgen = X1xA共1兲 + X2xA共2兲 + X3xA共3兲
xBgen = X1xB共1兲 + X2xB共2兲 + X3xB共3兲
gen
xMg
=
共1兲
X1xMg
+
共2兲
X2xMg
+
共3兲
X3xMg
,
冧
∆
共1兲 共1兲
关xA , T兴
Gat
共14兲
where the second “共2兲” and third “共3兲” phases are at stoichiometry. In addition, we have
nuc
共1兲
共1 − xA共i兲关xA共1兲,T兴兲,
nuc + muc
共1兲
xMg
= 1 − xA共1兲 − xB共1兲 ,
共15兲
where xA共1兲共i兲关xA共1兲 , T兴 is the equilibrium concentration of A interstitials in the first phase given by Eqs. 共8兲 and 共9兲 at the
chosen total concentration xA共1兲. The minimization of Gmix
at
with respect to xA共1兲 gives the solubility xA共1兲共T兲 in the first
phase “共1兲”:
共1兲
mix 共1兲
Gmix
at 关xA 共T兲,T兴 = min Gat 关xA ,T兴.
共1兲
xA
∆
xB共1兲 =
共16兲
This procedure, equivalent to the common-tangent method,
is the generalization of the approach developed in Ref. 24 to
the case of ternary alloys. The method is somehow different
from that developed in Ref. 28. Although, the regular solution model used in Ref. 28 and the presented ideal solution
model coincides in the low-solubility regime, the consideration of only two ground states 共“gs”兲 by the authors of Ref.
28, differs from our approach requiring the knowledge of the
whole ternary stability. This is because the disordered
Mg-B-A solution does not generally belong to the
Mg-B↔ gs共2兲 or Mg-B↔ gs共3兲 lines in the ternary phase diagram. Thus, we have to consider the phase mixture of
MgnucBmuc, gs共2兲, and, gs共3兲 to guarantee accurate estimation
of the solubility.
FIG. 1. 共Color online兲 Formation energies ⌬Ebin
at for Mg-B, B-A,
and Mg-A binary ground states 共A = Li, Be, Na, Mg, K, Ca, Rb, Sr,
Cs, Ba兲 from Table IV. The dashed parabola for the Li-B system
indicates the linear chains phases of Ref. 32.
end of section兲. Hence, from Eqs. 共13兲 and 共16兲 we obtain
共1兲
⳵ G共1兲
⳵ Gmix
⳵ Emix
at 关xA ,T兴
at
at
⬇
+
X
= 0,
1
⳵ xA共1兲
⳵ xA共1兲
⳵ xA共1兲
共17兲
共1兲 gen
共1兲
共2兲
共3兲
Emix
at 关xA ,x 兴 = X1Eat + X2Eat + X3Eat ,
共18兲
where
A. Low-solubility approximation
In order to get the analytical expression for equilibrium
solubilities from Eq. 共16兲, further approximations are required: 共a兲 the equilibrium concentration of solute A is small
and 共b兲 only substitutional or interstitial positions of one type
␣ are occupied 共this approximation will be eliminated at the
where the fractions Xk = Xk关xA共1兲 , xgen兴 are determined from
Eqs. 共14兲 and 共15兲. Note that the ca. 共⬇兲 in Eq. 共17兲 corresponds to approximation 共a兲 applied to Eqs. 共7兲 and 共10兲,
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PHYSICAL REVIEW B 79, 134203 共2009兲
ROMAN V. CHEPULSKII AND STEFANO CURTAROLO
⳵ X1 共1兲 ⳵ X1 共1兲
Gat ⬇ 共1兲 Eat .
⳵ xA共1兲
⳵ xA
共19兲
Equations 共14兲 and 共15兲 lead to
⳵ Xk
⳵ Xk
,
共1兲 ⬇ − X1
⳵ xAgen
⳵ xA
共20兲
and through Eq. 共18兲,
⳵ Emix
⳵ Emix
at
at
⬇
−
X
.
1
⳵ xAgen
⳵ xA共1兲
共21兲
By using the expression of G共1兲 from Eq. 共7兲, we have
⳵ G共1兲
c␣
at
⬇ ␮␣ + kBT ln
,
1 − c␣
⳵ xA共1兲
共22兲
where the ca. 共⬇兲 corresponds to approximation 共a兲.29
Equation 共17兲 can be solved with respect to c␣ with the
help of Eqs. 共21兲 and 共22兲 as
冋
c␣ = 1 + exp
␣
共A兲
Esol
k BT
册 冏 冉
−1
⯝ exp −
␣
共A兲
Esol
k BT
冊冏
␣ 共A兲
kBTⰆEsol
,
共23兲
where
␣
共A兲 = ␮␣ −
Esol
=
=
1
xAgen
1
xAgen
⳵ Emix
at
⳵ xAgen
冋
冉
gen
共1兲
E共1兲
at + ␮␣xA − Eat +
冊册
⳵ Emix
at
xAgen
⳵ xAgen
mix gen
gen
关E共1兲
at 共x 兲 − Eat 共x 兲兴.
共24兲
Both Eqs. 共3兲 and 共24兲 define the same quantity: the low␣
共A兲, determining the low
solubility formation energy, Esol
solubility of A solutes in ␣-type positions. In addition, by
using Eq. 共8兲 共Ref. 29兲 the total equilibrium concentration of
solute A in the phase 共1兲 becomes
xA共1兲 ⬇
␯c␣
,
nuc + muc
共25兲
where c␣ is determined by Eq. 共23兲.
Approximation 共b兲 can be relaxed if the various types of
substitutional and interstitial positions are occupied independently. This is expected to be true in the low solute concentration limit. Thus, the expression 共25兲 can be integrated
through the various types of positions, leading to
xA共1兲 ⬇
兺 i ␯ ic i + 兺 s ␯ sc s
nuc + muc
,
共26兲
where ci and cs are obtained from a set of Eq. 共23兲, by using
共i兲
共s兲
共A兲 and Esol
共A兲 for each type of dethe corresponding Esol
fects.
FIG. 2. 共Color online兲 The ternary ground-state phase diagrams
of Mg-B-A alloys 共A = Li, Be, Na兲. The filled circles represent
known ground states. The blue stars and red triangles represent the
substitutional 共Mg7B56A, Mg3B16A, Mg7B16A兲 and interstitial
共Mg8B56A, Mg4B16A, Mg8B16A兲 supercells, respectively. Each supercell has been constructed from the most appropriate Mg-B compound 共MgB2, MgB4, or MgB7兲. The question mark in the MgB-Be system indicates the existence of new possible ternary ground
state共s兲 共Sec. VI兲.
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FIRST-PRINCIPLES SOLUBILITIES OF ALKALI AND…
FIG. 3. 共Color online兲 The ternary ground state phase diagrams
of Mg-B-A alloys 共A = K, Rb, Cs兲.
VI. FIRST-PRINCIPLES CALCULATIONS
The first-principles calculations of energies are performed
by using our high-throughput quantum calculations framework AFLOW 共Refs. 20 and 30–32兲 and the software
VASP.33 We use projector augmented waves 共PAW兲
pseudopotentials34 and exchange-correlation functionals as
FIG. 4. 共Color online兲 The ternary ground-state phase diagrams
of Mg-B-A alloys 共A = Ca, Sr, Ba兲.
parameterized by Perdew and Wang35 for the generalized
gradient approximation 共GGA兲. Simulations are carried out
without spin polarization 共not required for the elements under investigation兲, at zero temperature, and without zero-
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ROMAN V. CHEPULSKII AND STEFANO CURTAROLO
TABLE V. Interstitial 共i兲 and substitutional 共s兲 quantities for alkali and alkaline-earth solute elements A in
¯ s 关Eq. 共1兲兴, “true” formation energies
MgB2, MgB4, and MgB7 compounds: “raw” formation energies ␮i, ␮
共s兲
共i兲
共s兲
E共i兲
A , EA 关Eq. 共2兲兴 and low-solubility formation energies Esol, Esol 关Eq. 共3兲兴. In each case, the presented value
is the lowest one among all the possible interstitial or substitutional positions.
MgB4
MgB2
A
Li
Be
Na
K
Ca
Rb
Sr
Cs
Ba
共eV兲
¯s
␮i , ␮
E共i,s兲
A
共i,s兲
Esol
¯s
␮i , ␮
共i,s兲
EA
共i,s兲
Esol
¯s
␮i , ␮
共i,s兲
EA
共i,s兲
Esol
¯s
␮i , ␮
E共i,s兲
A
共i,s兲
Esol
¯s
␮i , ␮
E共i,s兲
A
共i,s兲
Esol
¯s
␮i , ␮
共i,s兲
EA
共i,s兲
Esol
¯s
␮i , ␮
E共i,s兲
A
共i,s兲
Esol
¯s
␮i , ␮
E共i,s兲
A
共i,s兲
Esol
¯s
␮i , ␮
E共i,s兲
A
共i,s兲
Esol
共i兲
0.542
2.444
2.716
−2.665
1.042
1.054
5.623
6.932
6.944
7.022
8.061
8.073
2.833
4.751
6.364
9.463
10.393
10.406
5.514
7.138
9.042
4.663
5.523
5.536
2.450
4.373
5.971
共s兲
−0.111
0.310
0.574
−0.876
1.350
1.21
1.839
1.666
1.526
4.674
4.232
4.092
−0.295
−0.732
1.747
6.572
6.021
5.981
1.761
1.618
3.802
8.436
7.815
7.675
3.959
3.518
5.99
共i兲
0.234
2.136
2.654
−0.506
3.201
3.201
2.409
3.718
3.718
2.984
4.023
4.023
1.042
2.960
4.993
3.733
4.663
4.862
1.334
2.958
5.282
4.559
5.419
5.419
1.365
3.288
5.306
MgB7
共s兲
−0.109
0.312
0.686
−0.710
1.516
1.256
1.285
1.113
0.802
3.001
2.559
2.248
−1.107
−0.671
1.218
3.957
3.406
3.460
0.221
0.364
2.544
5.064
4.444
4.134
1.084
1.525
3.400
共i兲
−0.959
0.943
1.657
−1.577
2.130
2.181
1.999
3.308
3.308
3.453
4.492
4.492
1.251
3.169
5.453
4.751
5.681
6.047
2.149
3.773
6.348
6.488
7.349
7.349
3.118
5.041
7.309
共s兲
−0.055
0.366
0.506
−2.100
0.126
−0.557
0.869
0.697
0.147
2.077
1.635
0.838
−1.941
−1.504
0.358
3.312
2.761
2.477
−0.960
−0.817
1.331
5.131
4.510
3.411
0.335
0.777
2.624
point motion. All structures are fully relaxed 共shape and volume of the cell and internal positions of the atoms兲. The
effect of lattice vibrations is omitted. Numerical convergence
to within about 1 meV/atom is ensured by enforcing a high
energy cutoff 共414 eV兲 and dense 4500 k-point meshes.
The existence of the ground states for a given binary A-B
system 共where A and B are two generic elements兲 is based on
binary bulk formation energy, which, for each phase ␾ with
stoichiometry AxABxB, is determined with respect to pure A
and B energies Eat共A兲 and Eat共B兲 as
A. Ground states determination
⌬Ebin
at 共␾兲 ⬅ Eat共␾兲 − xAEat共A兲 − xBEat共B兲.
The calculation of solubility of A in Mg-B compounds
requires the knowledge of the relevant ground states in the
ternary Mg-B-A system. The systems under investigations
have not been well characterized experimentally or theoretically, and only the binary Mg-B, B-A, and Mg-A systems
have been studied. Hence, we performed additional highthroughput searches to determine if further ground states exist in the three binary systems.20,30–32 Based on the knowledge of the binary systems, we built36 the ternary ground
state phase diagrams for Mg-B-A, with the expectation that
no missed ternary ground state is relevant to the solubility of
A 共Ref. 37兲.
The results are presented at Table IV and in Fig. 1. For each
element A, the reference energy is chosen to be the lowest
among the pure fcc, bcc, and hcp energies.58 The reference
energy for boron is taken to be ␣-boron 共Refs. 19, 31, and
59–61兲.
All experimentally observed phases are confirmed except
for Sr9Mg38 共P63 / mmc兲. We also find two possible new
phases, CaB4 and RbB4, both P4 / mbm 共No. 127兲 and with
UB4 prototype 共CaB4 was previously identified in an ab initio online database51兲. The formation energies for Ba2Mg17,
Ba6Mg23, BaMg2, and CaMg2 are similar to those of Ref. 62
共reported as −0.079, −0.089, −0.088, and −0.126 eV, respec-
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PHYSICAL REVIEW B 79, 134203 共2009兲
FIRST-PRINCIPLES SOLUBILITIES OF ALKALI AND…
FIG. 5. 共Color online兲 Representation of the formations energies
from Table V. Dashed 共green兲 and dotted 共red兲 lines connect the
lowest formation energies for alkali and alkaline-earth elements,
respectively.
tively兲. The small differences can be explained on the basis
of different GGA pseudopotentials 共PW91 versus PBE兲 and
different energy cutoffs 共414 versus 360 eV兲. Our results
cannot be compared with the thermodynamic discussion
based on the semiempirical Miedema method reported in
Ref. 14, because we include additional ground-state prototypes other than those used in the heat of formations fitting in
Ref. 14.
The calculated ternary ground-state phase diagrams for
Mg-B-A alloys are depicted in Figs. 2–4. Note that in each
phase diagram, the triangles and stars represent the supercells with one interstitial or substitutional A atom, respectively. In the interstitial case, the supercells belong to the
lines MgnBm ↔ A, while in the substitutional case, the supercells belong to the lines parallel to Mg↔ A and intersecting
the MgnBm 共constant B concentration兲.
B. Formation energies numerical results
¯ s 关Eq.
The calculated parameters of the model 共“raw” ␮i, ␮
共i兲
共s兲
共i兲
共s兲
共1兲兴, “true”-EA , EA 关Eq. 共2兲兴 and low-solubility Esol, Esol
关Eq. 共3兲兴 formation energies兲 are presented in Table V and in
Fig. 5.
The values of the low-solubility energies reported in
共s兲
for substituTable V suggest the following. 共a兲 Only Esol
tional Be in MgB7 was found to be negative, indicating the
existence of unknown ground state共s兲 within the triangle
MgB7 ↔ Be3B50 ↔ Be1.11B3. This fact is summarized by the
question mark in Fig. 2. 共b兲 Generally, the substitutional systems have lower formation energy than the interstitial sys-
tems. Exceptions are Be, Cs, and Ba in MgB2; 共c兲 although
oscillating, the formation energies tend to increase with the
element number, as shown in Fig. 5. Regular oscillations are
observed for all substitutionals systems starting with Na. The
lower and higher boundaries of such correspond to alkaline
earth and alkali elements, respectively. 共d兲 Na, Ca, and Li
substitutionals in MgB7 have the lowest formation energies
共0.147, 0.358, and 0.506 eV, respectively兲, resulting in high
solubility 共Sec. VII兲. 共e兲 In general for substitutionals systems, the higher boron contents the lower formation energy
共except Li and Be兲. 共f兲 The corresponding true EA共i,s兲 and low共i,s兲
共A兲 formation energies generally demonstrate
solubility Esol
a similar behavior as functions of A. Exceptions are EA共s兲
共s兲
⬍ 0 for A = Ca and Sr and Esol
共A兲 ⬍ 0 for A = Be. The difference is due to the consideration of all the known ground
共i,s兲
and not only the pure Mg,
states in the determination of Esol
共i,s兲
B and A references 共as for EA 兲.
In Ref. 17, for Li-Mg-B the formation energies were obtained from first principles for interstitial and substitutional
Li in boron-rich MgB2 共3.00 and 0.27 eV, respectively兲.
These results are different form our low-solubility formation
共i兲
共s兲
共Li兲 = 2.72 eV and Esol
共Li兲 = 0.57 eV 共see Table
energies Esol
V兲, because only the hexagonal ␣-LiB phase was considered
as a ground state in Ref. 17 and the other boron-rich phases
were reported to be unstable 共the ␣-LiB was refined in Ref.
32 by variational minimization of the Li and B concentrations as Li8B7兲. However, in our high-throughput framework,
we found LiB3 and Li3B14 to be stable,63 and the appropriate
analysis of the formation energies should be done with respect to the triangle MgB2 ↔ LiB3 ↔ Mg as shown in Fig. 2.
In the low-solubility limit of Li, since the supercell composition is close to MgB2, the lever rule of phase decomposition of the mixtures does not cause significant errors if the
energies are calculated with respect to other references, and
therefore the results of Ref. 17 are somehow similar.
For Na-Mg-B, it can be shown that the substitutional Na
formation energy in magnesium-rich MgB2 共1.74 eV兲 obtained in Ref. 17 is equivalent to our “true” formation energy
共s兲
= 1.67 eV, see Table V兲.
for substitutional Na in MgB2 共ENa
The observed small numerical difference between two results
can be attributed to the different energy cutoffs for the basis
set 共414 versus 312 eV兲.64
VII. SOLUBILITY RESULTS
Solubility results are presented in Table VI 共the lowsolubility approximation values are shown in brackets兲. We
report only values larger than 10−6 共Li, Na, Ca, and K兲. The
values on Table VI suggest the following. 共a兲 Na is the only
element which has a substantial solubility: Na in MgB7 is
⬃0.5– 1% at T = 650– 1000 K. 共b兲 The substitutional solubility of Be in MgB7 cannot be determined because the corre共s兲
关see Eq. 共3兲兴 is negative imsponding formation energy Esol
plying that the ground state list is not complete. Thus the
approach of Sec. V is not applicable. 共c兲 In agreement with
共s兲
共Table V and Fig. 5兲,
their lowest formation energies Esol
substitutional Li, Na, and Ca experience the largest solubilities among all studied systems; 共d兲 due to their considerably
共i兲
共Table V and Fig. 5兲, all the
high formation energies Esol
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PHYSICAL REVIEW B 79, 134203 共2009兲
ROMAN V. CHEPULSKII AND STEFANO CURTAROLO
共1兲
共1兲
TABLE VI. Calculated equilibrium interstitial xA共i兲 共i兲 and substitutional xA共s兲 共s兲 A-solute concentrations
at temperatures T = 300, 650, and 1000 K, in MgB2, MgB4, and MgB7 compounds. The values obtained
within the low-solubility approximation are reported in brackets. Zeros are used for values smaller than 10−6.
共s兲
For Be “⌬E ⬍ 0” indicates that the corresponding formation energy Esol
关see Eq. 共3兲兴 is negative implying that
the ground state共s兲 list is not complete and the approach of Sec. V is not applicable.
共1兲
MgB2
xA共i,s兲
A
Li
Be
Na
K
Ca
Rb
Sr
Cs
Ba
T
共K兲
300
650
共i兲
0
0
1000
0
300
650
1000
300
MgB4
MgB7
共i兲
0
0
共s兲
0
0
共i兲
0
0
0
0
0
0
0
6.9⫻ 10−5
共7.0⫻ 10−5兲
0
0
0
0
0
0
0
0
0
共s兲
0
1.3⫻ 10−5
共1.2⫻ 10−5兲
4.5⫻ 10−4
共4.3⫻ 10−4兲
0
0
0
0
0
0
0
0
650
0
0
0
0
0
1000
0
0
0
0
300
650
1000
300
650
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1.8⫻ 10−5
共1.8⫻ 10−5兲
0
0
0
0
0
0
0
0
0
0
1000
0
0
0
0
0
300
650
1000
300
650
1000
300
650
1000
300
650
1000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
investigated interstitial systems have negligible solubilities.
共e兲 The low-solubility approximation and the general theory
agree within 5%.
The calculated negligible solubilities of Be, Na, Ca
agree
with
previous
experimental
in
MgB2
observations.4,7,8,11–13 We did not find any sign of nonvanishing solubilities of Rb, Cs, and Ba in the MgB2 alloy even at
high annealing temperature in disagreement with the reported experimental data of Ref. 9. However, our negligible
Rb and Cs solubility in bulk MgB2 are in agreement with the
共s兲
0
9.9⫻ 10−6
共9.9⫻ 10−6兲
2.6⫻ 10−4
共2.6⫻ 10−4兲
⌬E ⬍ 0
⌬E ⬍ 0
⌬E ⬍ 0
2.1⫻ 10−4
共2.1⫻ 10−4兲
4.4⫻ 10−3
共4.2⫻ 10−3兲
1.0⫻ 10−2
共0.96⫻ 10−2兲
0
0
3.6⫻ 10−6
0
1.1⫻ 10−4
共1.0⫻ 10−4兲
1.0⫻ 10−3
共0.96⫻ 10−3兲
0
0
0
0
0
0
0
0
0
0
0
0
experimental conclusion made in Ref. 10 where the authors
suggest that Rb and Cs dopants most likely segregate in
grain boundaries.
The obtained low solubilities of Li and Sr in MgB2 differ
from experimental vales 共solubility up to 30% in Refs. 2–6
and 13兲. This discrepancy can be attributed to segregation of
Li and Sr in the grain boundaries as was concluded for Rb
共s兲
= 3.8 eV for
and Cs in Ref. 10. In particular, the value of Esol
Sr is too large for non-negligible bulk solubility. For Li in
共s兲
= 0.57 eV is not
MgB2 the calculated formation energy Esol
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PHYSICAL REVIEW B 79, 134203 共2009兲
FIRST-PRINCIPLES SOLUBILITIES OF ALKALI AND…
very large and it is the smallest among the investigated solutes in MgB2. In Ref. 17, qualitative conclusions about high
solubility of Li and low solubility of Na in MgB2 were made
based on a large difference between the corresponding formation energies. Our numerical results demonstrate that, despite much lower formation energy of Li in comparison with
Na in MgB2, the Li solubility is still very low 共Sec. VI兲.
It should be emphasized that we consider the alloy in
thermodynamical equilibrium which can be difficult to reach
at low temperatures and the experimental equilibrium solubility tends usually to be overestimated. In fact, the formation of metastable and/or unstable states which are subsequently frozen at low temperatures, can make solubility
measurements very challenging. In such scenarios, the measured solubility may correspond to spinodal concentration
rather then actual binodal concentration or simply characterize the frozen out of equilibrium solution remaining from the
initial specimen preparation at higher temperature. Besides,
the segregation of defects into grain boundaries, especially in
multicrystalline samples prepared through nonoptimal cooling, dramatically affects the amount of frozen defects and
solutes.
Although in our study we did not perform an extensive
search over the configurational space of ternary alloy, if a
new ternary phases were present near MgB2, the solute atoms would concentrate and nucleate such phases, and this
may be misinterpreted as a high solubility in MgB2 phase.
Furthermore, a new equilibrium ternary phase would result
in an increase in impurities formation energies and, correspondingly, in a decrease in solubility. Numerical approximations in the first principles are not expected to affect the
values of solubility: i.e., typically for ␦Esol ⬃ 30 meV/ atom,
␦xA共1兲 ⬃ 2 ⫻ 10−4. Lattice vibrations and solute-solute interactions are neglected because they are not expected to play
important roles at low temperatures and low solute concentrations.
VIII. CONCLUSIONS
In the present paper, we present an approach to study
the solubilities in ternary alloys. The advantage of the ap-
*stefano@duke.edu
1 J.
Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J.
Akimitsu, Nature 共London兲 410, 63 共2001兲.
2 F. J. Owens, Physica C 363, 202 共2001兲.
3
Y. G. Zhao, X. P. Zhang, P. T. Qiao, H. T. Zhang, S. L. Jia, B. S.
Cao, M. H. Zhu, Z. H. Han, X. L. Wang, and B. L. Gu, Physica
C 361, 91 共2001兲.
4 X. P. Zhang, Y. G. Zhao, P. T. Qiao, Z. S. Yin, S. L. Jia, B. S.
Cao, M. H. Zhu, Z. H. Han, Y. H. Xiong, P. J. Li, and B. L. Gu,
J. Supercond. Novel Magn. 15, 159 共2002兲.
5 M. R. Cimberle, M. Novak, P. Manfrinetti, and A. Palenzona,
Supercond. Sci. Technol. 15, 43 共2002兲.
6 J. Karpinski, N. D. Zhigadlo, S. Katrych, K. Rogacki, B. Batlogg, M. Tortello, and R. Puzniak, Phys. Rev. B 77, 214507
共2008兲.
7
P. Toulemonde, N. Musolino, H. L. Suo, and R. Flükiger, J.
proach consists in considering all the known ternary ground
states rather than just the pure references. Furthermore, we
propose an analytical low-solubility approximation that can
be used for high-throughput calculations of solubilities in
alloys.
Combining the developed framework with first-principles
calculations, we have determined the formation energies and
solubilities of alkali 共Li, Na, K, Rb, Cs兲 and alkaline earth
共Be, Ca, Sr, Ba兲 metals in the Mg-B system. It is found that
the considered metals have low solubilities in the boron-rich
Mg-B alloy. Substitutional Na, Ca, and Li experience the
largest solubilities, with Na in MgB7 reaching 0.5%–1% at
T = 650– 1000 K. All the considered interstitial scenarios
lead to negligible solubilities. The solubility of Be in MgB7
cannot be determined within our model because the corresponding low-solubility formation energy is negative implying that the existing ground states list must be augmented
through a more extensive search over the configurational
space.
We have also presented a high-throughput search of
ground states in binary Mg-B, Mg-A, and B-A alloys 共A
= Li, Be, Na, Mg, K, Ca, Rb, Sr, Cs, Ba兲. Ternary phase
diagrams Mg-B-A are constructed based on of the determined phases. Sr9Mg38 is not an equilibrium ground state
despite its high temperature validations. Two new ground
states CaB4 and RbB4 are found.
ACKNOWLEDGMENTS
We acknowledge Mike Mehl, Igor Mazin, and Wahyu Setyawan for fruitful discussions. This research was supported
by ONR 共Grants No. N00014-07-1-0878 and No. N0001407-1-1085兲 and NSF 共Grant No. DMR-0639822兲. We thank
the Teragrid Partnership 共Texas Advanced Computing Center, TACC兲 for computational support 共Grant No. MCA07S005兲.
Supercond. Novel Magn. 15, 613 共2002兲.
Agostino, M. Panetta, P. Volpe, M. Truccato, S. Cagliero, L.
Gozzelino, R. Gerbaldo, G. Ghigo, F. Laviano, G. Lopardo, and
B. Minetti, IEEE Trans. Appl. Supercond. 17, 2774 共2007兲.
9
A. V. Palnichenko, O. M. Vyaselev, and N. S. Sidorov, JETP
Lett. 86, 272 共2007兲.
10
R. K. Singh, Y. Shen, R. Gandikota, D. Wright, C. Carvalho, J.
M. Rowell, and N. Newman, Supercond. Sci. Technol. 21,
025012 共2008兲.
11
I. Felner, Physica C 353, 11 共2001兲.
12 C. H. Cheng, Y. Zhao, X. T. Zhu, J. Nowotny, C. C. Sorrell, T.
Finlayson, and H. Zhang, Physica C 386, 588 共2003兲.
13 A. Tampieri, G. Celotti, S. Sprio, and D. Rinaldi, Int. J. Mod.
Phys. B 17, 438 共2003兲.
14 X. S. Wu and J. Gao, Physica C 418, 151 共2005兲.
15
A. R. Miedema, P. F. de Chatel, and F. R. de Boer, Physica B &
8 A.
134203-11
PHYSICAL REVIEW B 79, 134203 共2009兲
ROMAN V. CHEPULSKII AND STEFANO CURTAROLO
C 100, 1 共1980兲.
Toop, Trans. Metall. Soc. AIME 237, 738 共1965兲.
17
F. Bernardini and S. Massidda, Europhys. Lett. 76, 491 共2006兲.
18 T. B. Massalski, Binary Alloy Phase Diagrams, 2nd ed. 共ASM
International, Materials Park, OH, 2003兲.
19
Pauling File: Inorganic Materials Database and Design
System—Binaries Edition, edited by P. Villars 共ASM International, Metals Park, OH, 2002兲.
20 S. Curtarolo, D. Morgan, and G. Ceder, CALPHAD: Comput.
Coupling Phase Diagrams Thermochem. 29, 163 共2005兲.
21 S. Curtarolo, AFLOW: a computational software to calculate properties of materials in a high-throughput fashion. http://
materials.duke.edu/aflow.html
22
Y. Mishin and D. Farkas, Philos. Mag. A 75, 169 共1997兲; Y.
Mishin and C. Herzig, Acta Mater. 48, 589 共2000兲.
23
¯ s indicates that the quantity is not a chemical potenThe bar at ␮
tial itself, as the number of particle is conserved.
24 R. V. Chepulskii and S. Curtarolo, arXiv:0901.0200 共unpublished兲.
25 M. A. Krivoglaz and A. A. Smirnov, The Theory of OrderDisorder in Alloys 共Macdonald, London, 1964兲.
26 C. Wagner and W. Schottky, Z. Phys. Chem. Abt. B 11, 163
共1930兲.
27 In this method, the general composition xgen plays just a supplementary role not effecting the results if xgen necessarily belongs
to the three-phase compositional triangle.
28
C. Sigli, L. Maenner, C. Sztur, and R. Shahni, in Proceedings of
the 6th International Conference on Aluminum Alloys (ICAA-6),
edited by T. Sato et al. 共Japan Institute of Light Metals, 1998兲,
Vol. 1, p. 87.
29 In this case, the approximation 共a兲 implies the neglection of x共1兲
A共i兲
in Eq. 共15兲 and ci in the denominator of Eq. 共10兲, respectively.
30
S. Curtarolo, D. Morgan, K. Persson, J. Rodgers, and G. Ceder,
Phys. Rev. Lett. 91, 135503 共2003兲.
31 A. N. Kolmogorov and S. Curtarolo, Phys. Rev. B 73,
180501共R兲 共2006兲.
32 A. N. Kolmogorov and S. Curtarolo, Phys. Rev. B 74, 224507
共2006兲.
33 G. Kresse and J. Hafner, Phys. Rev. B 47, 558 共1993兲.
34 P. E. Blochl, Phys. Rev. B 50, 17953 共1994兲.
35 Ailan Cheng, Michael L. Klein, and Laurent J. Lewis, Phys. Rev.
B 44, 13 共1991兲.
36
http://www.qhull.org/.
37 The approximation implies that a true-ternary ground state does
not exist or, if it exist, it is in a region of the phase diagram not
relevant for solubility of A in MgB2, MgB4, and MgB7.
38 International Tables for Crystallography, edited by T. Hahn 共D.
Reidel, Dordrecht, 1983兲, Vol. A.
39 G. Mair, R. Nesper, and H. G. von Schnering, J. Solid State
Chem. 75, 30 共1988兲.
40
G. Mair, H. G. von Schnering, M. Wörle, and R. Nesper, Z.
16 W.
Anorg. Allg. Chem. 625, 1207 共1999兲.
X. Zhang, F. F. Xu, and T. Tanaka, J. Solid State Chem. 177,
3070 共2004兲.
42 J. Y. Chan, F. R. Fronczek, D. P. Young, J. F. DiTusa, and P. W.
Adams, J. Solid State Chem. 163, 385 共2002兲.
43
T. W. Baker, Acta Crystallogr. 15, 175 共1962兲.
44
R. Naslain and J. S. Kasper, J. Solid State Chem. 1, 150 共1970兲.
45 B. Albert and K. Hofmann, Z. Anorg. Allg. Chem. 625, 709
共1999兲.
46 A. Guette, M. Barret, R. Naslain, P. Hagenmuller, L. E. Tergenius, and T. Lundström, J. Less-Common Met. 82, 325 共1981兲.
47
A. Guette, R. Naslain, and J. Galy, C. R. Hebd. Seances Acad.
Sci., Ser. A B, Sci. Math. Sci. Phys 275, 41 共1972兲.
48
N. V. Vekshina, L. Y. Markovskii, Y. D. Kondrashev, and T. K.
Voevodskaya, J. Appl. Chem. USSR 44, 970 共1971兲.
49
R. Naslain and J. Etourneau, C. R. Hebd. Seances Acad. Sci.,
Ser. A B, Sci. Math. Sci. Phys 263, 484 共1966兲.
50 L. Pauling and S. Weinbaum, Z. Kristallogr. 87, 181 共1934兲.
51
M. Mihalkovic and M. Widom, Alloy Database, http://
alloy.phys.cmu.edu/.
52 H. Nowotny, Z. Metallkd. 37, 31 共1946兲.
53 P. P. Blum and F. Bertaut, Acta Crystallogr. 7, 81 共1954兲.
54
E. I. Gladyshevskii, P. I. Kripyakevich, M. Y. Teslyuk, O. S.
Zarechnyuk, and Yu. B. Kuz’ma, Sov. Phys. Crystallogr. 6, 207
共1961兲.
55 F. E. Wang, F. A. Kanda, C. F. Miskell, and A. J. King, Acta
Crystallogr. 18, 24 共1965兲.
56 E. I. Gladyshevskii, P. I. Kripyakevich, Yu. B. Kuz’ma, and M.
Y. Teslyuk, Sov. Phys. Crystallogr. 6, 615 共1961兲.
57
E. Hellner and F. Laves, Z. Kristallogr. 105, 134 共1943兲.
58 Y. Wang, S. Curtarolo, C. Jiang, R. Arroyave, T. Wang, G. Ceder,
L. Q. Chen, and Z. K. Liu, CALPHAD: Comput. Coupling
Phase Diagrams Thermochem. 28, 79 共2004兲.
59
Z. Liu, X. Qu, B. Huang, and Z. Li, J. Alloys Compd. 311, 256
共2000兲.
60
H. Rosner and W. E. Pickett, Phys. Rev. B 67, 054104 共2003兲.
61 M. Widom and M. Mihalkovic, Phys. Rev. B 77, 064113 共2008兲.
62
H. Zhang, J. Saal, A. Saengdeejing, Y. Wang, L.-Q. Chen, and Z.
K. Liu, in Magnesium Technology 2007, edited by R. S. Beals,
A. A. Luo, N. R. Neelameggham, and M. O. Pekguleryuz 共Minerals, Metals and Materials Society/AIME, Warrendale, PA,
2007兲, p. 345.
63 H. B. Borgstedt and C. Guminsky, J. Phase Equilib. 24, 572
共2003兲.
64 As shown in Sec. V, the low-solubility formation energy
共s兲
共s兲
共Na兲 = 1.53 eV rather than true formation energy ENa
= 1.67
Esol
is more accurate for corresponding solubility characterization
when the system is miscible. The 0.14 eV difference between
共s兲
共s兲
and Esol
共Na兲 suggests that the effect of other ground state
ENa
共MgB4, see Fig. 2兲 on the formation energy and subsequently on
the solubility is not negligible.
41 F.
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