CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 173–182
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Calculation of phase equilibria for an alloy nanoparticle in contact with a solid nanowire Yann Eichhammer a,b,∗ , Marc Heyns b , Nele Moelans a a
Department of Material Science and Metallurgy, Katholieke Universiteit Leuven, Arenberg Kasteelpark 44, 3001-Heverlee, Belgium
b
IMEC, Kapeldreef 75, 3001-Heverlee, Belgium
article
info
Article history: Received 19 November 2010 Received in revised form 31 January 2011 Accepted 1 February 2011 Available online 11 March 2011 Keywords: Phase diagram Nanowire–Nanoparticle system Geometry Surface properties Interfaces
abstract Phase equilibria in a system constituted of an alloy nanoparticle in contact with a solid nanowire have been modelled based on the minimization of a Gibbs free energy function. The Gibbs free energy consists of a bulk, surface and interface contribution. The bulk contribution is taken from CALPHAD thermodynamic databases and the surface properties from the literature. The effect of particle size and surface and interfacial properties on the liquidus line of the Au–Ge and In–Si systems has been studied. The results are compared to the bulk phase diagram and phase equilibria calculated for nano-systems with different geometries. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction Currently, semiconducting nanowires are receiving considerable attention due to their potential applications in the manufacturing of electronic devices. They may potentially be used as channels in transistors presenting an all around gate architecture [1]. They also open new possibilities in the fabrication of solar cells [2] and their exceptional thermoelectric properties have been demonstrated [3]. Moreover, they might be used as sensing material for nano bio or chemical samples [4]. Several nanowire growth methods have been developed such as etching of bulk pieces of material [5], solution-based deposition [6], using molecular beams [7] and vapour phase deposition [8]. The last two methods are probably studied the most exhaustively since the first experimental demonstration of the vapour–liquid–solid mechanism by Wagner and Ellis [9]. In this mechanism, the vapour phase acts as a source for the nanowire material, which dissolves into a solid nanoparticle and forms a liquid alloy according to the phase diagram. When the liquid droplet is supersaturated, the nanowire element precipitates at the interface between the liquid droplet and the substrate.
∗ Corresponding author at: Department of Material Science and Metallurgy, Katholieke Universiteit Leuven, Arenberg Kasteelpark 44, 3001-Heverlee, Belgium. Tel.: +32 16321245; fax: +32 16321991. E-mail addresses:
[email protected] (Y. Eichhammer),
[email protected] (M. Heyns),
[email protected] (N. Moelans). 0364-5916/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2011.02.002
Thanks to extensive experimental studies, the principles of vapour–liquid–solid growth are known qualitatively but detailed quantitative descriptions of this process are still lacking and required to achieve a better control of nanowires growth. Several phenomena are involved in the vapour–liquid–solid growth and their relative importance depends on the growth setup. However, since we are dealing with nano-objects, size effects are always important. For nanoparticles, the depression of melting point has been studied experimentally [10] and theoretically [11–13] for pure and alloyed materials. Similar effects were experimentally demonstrated for systems constituted of a semiconducting nanowire and a metallic nanoparticle by Sutter and Sutter [14]. Earlier models have been proposed [15,16] to calculate equilibrium conditions between a nanowire and a nanoparticle. These models consider the influence of the Gibbs–Thomson effect and surface stress on the phase equilibria, but not the influence of the interface between the nanowire and the nanoparticle. In this paper, we propose as well to calculate phase equilibria for an alloy nanoparticle in contact with a solid nanowire, which is the situation of interest during nanowire growth even if growth itself can not be described as an equilibrium situation. Such calculations will yield important information on the state of the particle during growth experiments. Therefore, bulk Gibbs free energy expressions developed in the frame of CALPHAD [17,18] are modified to account for surface and interfacial effects. Moreover, we consider the composition dependence of the liquid surface energy as well as surface segregation, combining CALPHAD
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Fig. 1. General model description. The goal of the model is to predict the state of an alloy nanoparticle in contact with a solid nanowire, an important technologic parameter in nanowires growth processes. This can be represented as the liquidus line of the considered system.
Gibbs free energies and Butler’s equation [19,20]. CALPHAD-based approaches were developed before to calculate phase diagrams of nanoparticles. However, these methods only consider the effect of surfaces and not those of interfaces between two condensed phases, as present in a nanoparticle–nanowire system. We have also studied quantitatively for the different types of interfaces the importance of their effect on phase equilibria. The thermodynamic equations are developed in Section 2 and the numerical solution in Section 3. In Section 4, the calculated phase diagrams are given and the influence of model parameters is discussed. A schematic description of the model and of its implementation is given in Fig. 1.
supposed to have a circular cross-section, although real systems adopt faceted geometries, according to the Wulff construction. However, if the wire has a hexagonal cross-section, the surfaceto-volume ratio is
2. Thermodynamic model
Acircular /Vcircular =
We will consider a closed system constituted of a nanoparticle and a nanowire and calculate the Gibbs free energy of this system in two different states, namely the initial state constituted of a pure metallic solid nanoparticle and a pure semiconductor solid nanowire and the final state constituted of a liquid solution of metal and semiconductor and a pure semiconductor solid nanowire. 2.1. System geometry The geometry and the dimensions of the systems in the initial and the final state are depicted in detail in Figs. 2a–2c. Both in the initial and in the final state, the nanowire and the nanoparticle are
Ahexagonal /Vhexagonal =
6lwire ri √
3 3 2ri2 lwire
4
= √
3ri
(1)
where ri is the radius of the inner circle of the hexagon, whereas for a circular cross-section it is
=
2π rwire lwire
π rw2 ire lwire 2 rwire
.
(2)
Consequently, the assumption of a circular cross-section will not change the form of the equations but only the solid surface contributions. A geometric factor in solid surface energies can be used to account for a faceted geometry of the solid. The nanoparticle is assumed to be a truncated sphere, with a contact angle β on top of the nanowire. In the initial state, βi is determined based on experimental observation. In the final state, βf follows from the calculated volume of the liquid nanoparticle. The final contact angle is thus determined by geometric considerations and not with Young’s equation. We
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Fig. 2c. Zoom on the nanoparticle in the initial state. The particle is a hemisphere, of radius R0 part and the cap length is h0 . Rwire , R0part and h0 are linked by the relation h0 =
√
2 (R02 part − Rw ire ).
2.2. Thermodynamic equations Generally, the Gibbs free energy of small systems can be written as Gsystem = Gbulk + Gsurface . Fig. 2a. System in the initial state. The particle is a hemisphere of radius
R0part
and
the wire a cylinder of radius Rwire and of length l0wire .
(3)
Gbulk is the contribution from the bulk of the material and Gsurface is the contribution of the different surfaces or interfaces present in the system. The latter contribution is important for small systems due to their high surface-to-volume ratio. In principle, this method can be applied to any binary system. We have performed calculations for two different systems, Au–Ge and In–Si. A difference between both systems is the fact that the Si diamond phase can dissolve some In whereas the Ge diamond phase cannot dissolve Au. In this section, the equations are developed for the Au–Ge system. We will first develop the Gibbs free energy expressions of the system in the initial state. The nanoparticle in the initial state is pure metal in the solid state. Since the phase of the nanoparticle is the standard element reference phase of the metal as defined in the CALPHAD method, the bulk Gibbs free energy of this phase expressed with respect to this stable element reference phase is nanoparticle
G0
fcc H ,SER (J ) = npart ) 0 (GAu − GAu H ,SER = npart − GHAu,SER ) 0 (GAu
= 0.
(4)
The nanowire as well is in the standard element reference state. Therefore, ire ire Gw (J ) = nw (Gdiamond − GHGe,SER ) 0 0 Ge
= 0.
(5)
Generally, the Gibbs free energy of a surface or an interface, at constant temperature and pressure, can be written as follows: Gint = Aint γint Fig. 2b. System in the final state. The particle is a hemisphere of radius Rpart , the wire a cylinder of radius Rwire and of length lwire .
expect changes in the droplet shape in order to obey Young’s equation to have little effect on the liquid/vapour area and thus on the phase equilibria.
(6)
where Aint is the surface area of the interface and γint is the surface energy density of the interface. We have to consider the following interfaces, the particle–vapour interface, the wire–vapour interface and the particle–wire interface. The Gibbs free energy of the particle–vapour interface in the initial state equals part ,surf
G0
sol = (4π (R0part )2 − 2π R0part h0 )γAu .
(7)
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For a cylindrical wire, the Gibbs free energy of the wire–vapour interface equals w ire,surf
G0
sol = 2π Rwire l0wire γGe .
(8)
Finally, since the wire–particle interface is circular, the Gibbs free energy of this interface equals w ire/part ,inter
G0
sol,sol (J ) = π R2wire γAu ,Ge
(9)
sol sol where γAu and γGe are the surface energy density of solid Au sol and Ge, respectively, and γAu ,Ge the surface energy density of the interface between solid Au and Ge. The total Gibbs free energy of the system in the initial state is the sum of the bulk and the surface–interface contributions: system
Gi
= Gbulk + Gsurface i i sol = 0 + 0 + (4π (R0part )2 − 2π R0part h0 )γAu
(10)
Secondly, we will develop the expressions of the Gibbs free energy of the system in the final state. It is assumed that the wire has a constant radius, equal to its initial radius or in other words, the wire dissolves layer by layer. The liquid A–B solution is modelled as a substitutional solution. Therefore, the bulk Gibbs free energy of this solution is [17,18] liq
liq
liq xAu
where
H ,SER
liq
+ RT (
liq ln xAu
liq + xliq Au xGe
X
liq xAu
and
liq xGe
v
+
liq xGe
liq ln xGe
) !
liq
(11)
are the respective molar fractions of Au and H ,SER
liq
liq
H ,SER
Ge in the liquid solution, GAu − GAu and GGe − GGe are the lattice stabilities with respect to their reference state of liquid Au and Ge, respectively, R the universal gas constant, T the temperature. The last two terms of the above expression, respectively, represent the ideal mixing of Au and Ge elements and the deviation from the ideal mixing, which is modelled with a Redlich–Kister expansion [21], Lv being the interaction parameters of this Redlich–Kister expansion. The wire is assumed to remain pure Ge in its standard element reference phase. Thus, the Gibbs free energy of the wire remains 0 J. In the final state, two interfaces are modified; the particle–vapour interface has become a liquid–vapour interface and the wire–particle interface a solid–liquid interface. For the wire, only the surface area of the wire is modified. The surface energy of the particle, a liquid alloy, can be calculated as a function of its composition using Butler’s equation [20]:
γalloy = γAu +
RT AAu
surface
ln
xAu
xbulk Au
! +
1 AAu
,surface ,bulk (T , xsurface ) − Gex (T , xbulk × (Gex Au )) Au Au ! Au
= γGe +
RT
AGe
surface
ln
ex,surface GGe
×(
(T ,
xGe
xbulk Ge surface xGe
+
1
AGe
,bulk ) − Gex (T , xbulk Ge )) Ge
,bulk (T , xsurface ) = β Gex (T , xsurface ) (i = Au, Ge). i i i
(13)
The β coefficient is the ratio of the coordination number in the surface to the coordination number in the bulk. Tanaka et al. [22] propose a value of β = 0.83 for metallic liquid solutions. Molar surface can be linked to molar volume using the following equation [22]: 1
2
Ai = 1.091NA3 Vi 3
(14)
where NA is Avogadro’s number. Finally, the Gibbs free energy of the final state is the sum of the bulk and the surface and interface contributions: Gf
= Gbulk + Gsurface f f "
liq H ,SER = 0 + nliq xliq ) Au (GAu − GAu liq liq H ,SER + xliq ) + RT (xliq Ge (GGe − GGe Au ln xAu
# +
liq xGe
liq ln xGe
)+
liq liq xAu xGe
X v
liq Lv xAu
(
−
sol,liq sol + 2π Rwire lwire γGe + π R2wire γAu ,Ge .
Lv (xAu − xGe )v liq
ex,surface
liq xGe v
)
liq + (4π R2part − 2π Rpart h)γAu ,Ge
liq H ,SER ) + xliq ) Ge (GGe − GGe
xAu (GAu − GAu
ex,surface
Gi
system
sol,sol sol + 2π Rwire l0wire γGe + π R2wire γAu ,Ge .
GAu,Ge = nliq
ex,surface
of Au and Ge, respectively, and GAu and GGe the partial surface excess Gibbs free energy of Au and Ge, respectively. The latter can be linked to the partial bulk excess free energy of the surface as [22]
(12)
where γAu and γGe are the surface energy of pure liquid Au and Ge, respectively, AAu and AGe the molar surface area of pure liquid Au surface surface and Ge, respectively, xAu and xGe the surface molar fraction
(15)
3. Model implementation The different resolution steps will be described with more details here. It is explained first how the liquidus line is calculated and secondly how the transition temperature between the initial and the final state is determined. 3.1. Calculation of the liquidus line To find the position of the liquidus line on the Ge-rich side of the phase diagram, one has to compute the bulk equilibrium composition of the liquid nanoparticle as a function of temperature. As the transition temperature between the initial and the final state of the small system is not known beforehand, we need to perform this calculation on a broad temperature range, starting at a temperature lower than the bulk eutectic temperature. The minimization procedure is started at a temperature of 300 K, slightly above the lower limit of the validity range (298.15 K) of the thermodynamic functions and is repeated every temperature step up to a temperature of 1250 K, slightly above the bulk melting point of Ge or above the melting point of Si in the case of the In–Si system. For each temperature step, pressure, total number of moles of Au and Ge, final wire composition, final wire radius, initial wire length and initial particle radius are known whereas final particle radius, particle composition and final wire length are unknown. At each temperature, the liquid composition is varied between 0 and 1 with a given composition step and the unknown quantities are calculated as functions of this free variable as described below. Also the total Gibbs free energy of the system is calculated as a function of the free variable and finally minimization of the Gibbs free energy with respect to the free variable yields the equilibrium values of the unknowns. At each temperature and liquid composition point, we calculate the number of moles of each element in the liquid knowing that all
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177
Au is in the liquid phase: liq
liq
nAu = xAu nliq
(16)
liq
nAu
nliq =
(17)
liq xAu
liq
liq
nGe = xGe nliq
(18) liq
liq
where nliq is the number of moles of liquid and xAu and xGe the molar fraction of Au and Ge in the liquid, respectively. Knowing the number of moles in the liquid, one can calculate the volume of the liquid, assuming that the molar volume of the liquid is a linear combination of the molar volumes of pure liquids: liq
liq
liq
liq
liq
Vmol = xAu Vmol,Au + xGe Vmol,Ge Vliq =
liq nliq Vmol
(19)
.
(20)
As the particle is a truncated sphere, this liquid volume has to be equal to the volume of the truncated sphere and numerical resolution of the following equations yields the radius of the nanoparticle: Vliq −
4 3
π
R3part
q
h = Rpart −
+ πh
2
Rpart −
h 3
Fig. 3a. Calculated surface energy of an Au–Ge liquid alloy as a function of the bulk composition at a temperature of 773 K using Butler’s equation.
=0
(21)
R2part − R2wire .
(22)
These two equations are combined and solved using the iterative Newton–Raphson method yielding the two unknowns, the radius of the nanoparticle and the height of the spherical cap. Using Eq. (14), one can calculate the molar surface of the liquid, and, assuming that the surface is one monolayer thick, the number of moles at the surface: Asurf
nsurf =
surf
Amol
4π R2part − 2π Rwire h
=
surf Amol
.
(23)
Butler’s equation is now solved iteratively with the Newton–Raphson method to obtain the surface composition of the droplet. Since the element with the lowest surface energy, in this case Ge, has a tendency to segregate, the bulk composition of the droplet will be modified. We give in Figs. 3a and 3b the surface energy and the surface composition of an Au–Ge liquid solution as a function of the bulk composition, at a temperature of 773 K. The total number of moles of Au and Ge in the bulk of the droplet, accounting for segregation, equals liq
liq
surf
(24)
liq
liq
surf
(25)
nGe,bulk = nGe − nGe
nAu,bulk = nAu − nAu surf
surf
where nAu and nGe are the number of moles of Au and Ge at liq
liq
the surface as calculated with Butler’s equation. nGe,bulk and nAu,bulk liq
liq
yield a bulk liquid composition different from xGe and xAu , which is the free variable. Extra Ge has to be dissolved in the bulk to reach this initial composition:
Fig. 3b. Surface composition of an Au–Ge liquid alloy as a function of the bulk composition at a temperature of 773 K using Butler’s equation. Ge segregates to the surface due to its lower surface energy in the liquid phase.
The liquid quantity is calculated again, accounting for segregation in the liquid droplet and the previous calculations repeated to obtain the volume and the area of the liquid using Eqs. (17)– (20). Furthermore, knowing the amount of Ge in the liquid, the remaining amount of Ge in the wire is calculated from liq
nwire = nGe − nGe .
(27)
Therefore the volume of the wire is mol,sol
Vwire = nwire VGe
= π R2wire lwire .
(28)
As we made the assumption that the wire radius is unchanged, its new length and interfacial area are, respectively, Vwire /π R2wire and 2π Rwire lwire . Finally, the bulk part and the interfacial/surface part of the Gibbs free energy of the system in the final state are calculated as a function of the liquid composition and the total Gibbs free energy is minimized with respect to the liquid composition to obtain the equilibrium liquid composition.
liq
nGe,bulk liq
liq
nGe,bulk + nAu,bulk liq
liq nGe,bulk
=
3.2. Calculation of the transition temperature between the initial and final state
= xliq Ge (26)
liq
xGe nAu,bulk liq
1 − xGe
.
The transition temperature between the initial and the finale state cannot be called eutectic temperature since there are no three
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Table 1 Thermodynamic functions of the Au–Ge system (SSOL4 database). Standard element reference phase of gold 0,fcc 0,fcc GAu − HAu (298.15 K) = 298.15 < T < 929.40 − 6938.856 + 106.830098∗ T − 22.75455∗ T ∗ log(T ) − 0.00385924∗ T 2 + 3.79625∗ 10(−7)∗ T 3 − 25097∗ T (−1) 929.40 < T < 1337.33 − 93586.481 + 1021.69543∗ T − 155.706745∗ T ∗ log(T ) + 0.08756015∗ T 2 − 1.1518713∗ 10(−5)∗ T 3 + 10637210∗ T (−1) 1337.33 < T < 1735.80 + 314067.829 − 2016.37825∗ T + 263.252259∗ T ∗ log(T ) − 0.118216828∗ T 2 + 8.923844∗ 10(−6)∗ T 3 − 67999832∗ T (−1) 1735.80 < T < 3200 − 12133.783 + 165.272524∗ T − 30.9616∗ T ∗ log(T )
Gold liquid phase 0,liq 0,fcc GAu − HAu (298.15 K) = 298.15 < T < 929.40 5613.144 + 97.444232∗ T − 22.75455∗ T ∗ log(T ) − 0.00385924∗ T 2 + 3.79625∗ 10(−7)∗ T 3 − 25097∗ T (−1) 929.40 < T < 1337.33 − 81034.481 + 1012.30956∗ T − 155.706745∗ T ∗ log(T ) + 0.08756015∗ T 2 − 1.1518713∗ 10(−5)∗ T 3 + 10637210∗ T (−1) 1337.33 < T < 1735.80 + 326619.829 − 2025.76412∗ T + 263.252259∗ T ∗ log(T ) − 0.118216828∗ T 2 + 8.923844∗ 10(−6)∗ T 3 − 67999832∗ T (−1) 1735.80 < T < 3200 + 418.217 + 155.886658∗ T − 30.9616∗ T ∗ log(T ) Standard element reference phase of Germanium 0,dia 0,dia GGe − HGe (298.15 K) = 298.15 < T < 900 − 9486.153 + 165.635573∗ T − 29.5337682∗ T ∗ log(T ) + 0.005568297∗ T 2 − 1.513694∗ 10(−6)∗ T 3 + 163298∗ T (−1) 900 < T < 1211.40 − 5689.239 + 102.86087∗ T − 19.8536239∗ T ∗ log(T ) − 0.003672527∗ T 2 1211.40 < T < 3200 − 9548.204 + 156.708024∗ T − 27.6144∗ T ∗ log(T ) − 8.59809∗ 10(28)∗ T (−9) Germanium liquid phase 0,liq 0,dia GGe − HGe (298.15 K) = 298.15 < T < 900 + 27655.337 + 134.94853∗ T − 29.5337682∗ T ∗ log(T ) + 0.005568297∗ T 2 − 1.513694∗ 10(−6)∗ T 3 + 163298∗ T (−1) + 8.56632∗ 10(−21)∗ T 7 900 < T < 1211.40 + 31452.25 + 72.173826∗ T − 19.8536239∗ T ∗ log(T ) − 0.003672527∗ T 2 + 8.56632∗ 10(−21)∗ T 7 1211.40 < T < 3200 + 27243.473 + 126.324186∗ T − 27.6144∗ T ∗ log(T ) Interaction parameters of the liquid phase 0,liq LAu,Ge = −18059.75 − 13.08541∗ T 1,liq
LAu,Ge = −6131.6 − 9.10177∗ T 2,liq
LAu,Ge = −4733.85 − 3.25908∗ T 3,liq
LAu,Ge = −8120.5 − 5.82538∗ T
phases in equilibrium in our model. Furthermore, eutectic points may not exist for nano-systems [23]. However, the transition temperature of the nano-systems can be compared to the bulk eutectic temperature and the minimum semiconductor solubility in the liquid can be compared to the bulk eutectic composition. This comparison is relevant for the application to nanowires growth [24]. The transition temperature between the final and the initial state is obtained by comparing the Gibbs free energy of these two states. We calculate
1G =
system Gf
liq,equi xAu
(
, T) −
system Gi
(T )
(29)
liq,equi xAu
where G ( , T ) is the minimum Gibbs free energy of the final state at the temperature T . As long as 1G is positive, the initial state is the stable state, when it becomes negative, the final state is the stable state. final
4. Results All calculations were performed with an initial wire length of 300 nm. It is important to note that we have plotted liquidus lines as a function of the composition of the bulk of the droplet and not as a function of the averaged composition between the surface and the bulk of the droplet. 4.1. Comparison with the bulk phase diagram The liquidus line of the modified Au–Ge system is computed for the parameters listed in Tables 1 and 2 and wire radius rwire equal to 10 and 5 nm. The contact angle in the initial state is taken 90°. The values of the interfacial energies were chosen arbitrarily for lack of information in the literature. We chose values of the same order of magnitude than the surface energies, assuming a smaller value of the solid–liquid interfacial energy compared to the solid–solid interfacial energy because of elastic contributions to the latter value. These two liquidus lines are compared to the
Table 2 Thermophysical properties in the Au–Ge system. sol γAu
1.410 J/m2
sol γGe
1.32 J/m2
γ
liq Au
[25] [26]
1.169 − 2.5 10 ∗
(−4)∗
(T −
melting TAu
liq γGe
0.621 − 2.6∗ 10(−4)∗ (T − TGe
liq,sol γAu ,Ge
0.05 J/m2
sol,sol γAu ,Ge
1 J/m2
sol Vmol ,Au
1.0255∗ 10(−5)
sol Vmol ,Ge
1.36464∗ 10(−5)
liq
Vmol,Au liq Vmol,Ge
melting
)
[27]
)
[27]
1.13∗ 10(−5)∗ (1 + 0.69∗ 10(−4)∗ (T − TAu
melting
(−5)∗
1.320
(1 + 0.8 10 ∗
(−4)∗
(T −
melting TGe
))
))
[27] [27]
bulk liquidus line in Fig. 4. The transition temperature is lower in nano-systems compared to the bulk eutectic temperature. It equals 523 K in the 10 nm system and 416 K in the 5 nm system whereas the bulk eutectic temperature equals 629 K. The minimum Ge solubility in the droplet also decreases, namely from 29 at.% Ge at the eutectic point for the bulk system to 24 at.% Ge slightly above the transition temperature in the 10 nm system and to 20 at.% Ge in the 5 nm system. One can also note that at a given temperature, the Ge solubility in the liquid is more important in nano-systems than in the bulk system. Similar calculations are carried out for the In–Si system. However the contact angle was taken equal to 115° for this system, following experimental observation [8]. The calculated liquidus lines for wire radii of 10 and 5 nm are given in Fig. 5. The diagrams have been calculated using the parameters listed in Tables 3 and 4. The transition temperature in the nano-systems is also lower than the bulk eutectic temperature. It equals 322 K in the 10 nm system whereas the bulk eutectic temperature equals 429.75 K. In the 5 nm system, this temperature is below 298.15 K, the lowest
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Table 3 Thermodynamic functions of the In–Si system. Standard element reference phase of indium 0,BCT −A6
0,BCT −A6
GIn − HIn (298.15 K) = 298.15 < T < 429.75 − 6978.89 + 92.338115∗ T − 21.8386∗ T ∗ log(T ) − 0.00572566∗ T 2 − (2.120321e − 06)∗ T 3 − 22906∗ T (−1) 429.75 < T < 3800 − 7033.516 + 124.476588∗ T − 27.4562∗ T ∗ log(T ) + (5.4607e − 04)∗ T 2 − (8.367e − 08)∗ T 3 − 211708∗ T (−1) + (3.53116e + 22)∗ T (−9) Indium liquid phase 0,liq 0,liq GIn − HIn (298.15) = 298.15 < T < 429.75 − 3696.798 + 84.701255∗ T − 21.8386∗ T ∗ log(T ) − 0.00572566∗ T 2 − (2.120321e − 06)∗ T 3 − 22906∗ T (−1) − (5.59058e − 20)∗ T 7 429.75 < T < 3800 − 3749.81 + 116.835784∗ T − 27.4562∗ T ∗ log(T ) + (5.4607e − 04)∗ T 2 − (8.367e − 08)∗ T 3 − 211708∗ T (−1) Indium diamond phase 0,liq 0,dia GIn − HIn (298.15) = 298.15 < T < 429.75 − 2794.89 + 92.338115∗ T − 21.8386∗ T ∗ log(T ) − 0.00572566∗ T 2 − (2.120321e − 06)∗ T 3 − 22906∗ T (−1) 429.75 < T < 3800 − 2849.516 + 124.476588∗ T − 27.4562∗ T ∗ log(T ) + (5.4607e − 04)∗ T 2 − (8.367e − 08)∗ T 3 − 211708∗ T (−1) + (3.53116e + 22)∗ T (−9) Standard element reference phase of Silicon 0,dia 0,dia GSi − HSi (298.15 K) = 298.15 < T < 1687 − 8162.609 + 137.236859∗ T − 22.8317533∗ T ∗ log(T ) − 0.001912904∗ T 2 − (3.552e − 09)∗ T 3 + 176667∗ T (−1) 1687 < T < 3800 − 9457.642 + 167.281367∗ T − 27.196∗ T ∗ log(T ) − (4.20369e + 30)∗ T (−9) Silicon liquid phase 0,liq 0,liq GSi − HSi (298.15 K) = 298.15 < T < 1687 + 42533.751 + 107.13742∗ T − 22.8317533∗ T ∗ log(T ) − 0.001912904∗ T 2 − (3.552e − 09)∗ T 3 + 176667∗ T (−1) + (2.09307e − 21)∗ T 7 1687 < T < 3800 + 40370.523 + 137.722298∗ T − 27.196∗ T ∗ log(T ) Interaction parameters of the liquid phase 0 ,L LIn,Si = +45100 − 12.8∗ T Interaction parameters of the diamond phase 0,dia LIn,Si = 50∗ T
Fig. 4. Calculated liquidus line of the Ge-rich side of the Au–Ge system for different wire radii.
Table 4 Thermophysical properties in the In–Si system.
γInsol
0.633 J/m2
γ
1.510 J/m2
sol Si
[25] [28]
γInliq
0.556 − 0.09∗ 10(−3)∗ (T − TIn
)
[27]
γSiliq
0.865 − 0.13∗ 10(−3)∗ (T − TSi
)
[27]
γInliq,Si,sol
0.05 J/m2
γInsol,Si,sol
1 J/m2
sol Vmol ,In
1.57∗ 10(−5)
sol Vmol ,Si
1.2∗ 10(−5)
melting melting
Vmol,In
1.63∗ 10(−5)∗ (1 + 9.7∗ 10(−5)∗ (T − TIn
liq Vmol,Si
1.11 10
liq
melting
∗
(−5)∗
(1 + 1.4 10 ∗
(−4)∗
))
[27]
(T − TSimelting ))
[27]
Fig. 5. Calculated liquidus line of the Si-rich side of the In–Si system for different wire radii.
temperature at which bulk Gibbs free energy expressions are valid. Consequently, an exact value cannot be given in the 5 nm system. In general, it is not possible to calculate the minimum Si solubility in the nano-systems due to the extremely low Si solubility at the eutectic point, which cannot be treated numerically. 4.2. Comparison of different geometries The results of the new model are here compared to the results obtained for spherical nanoparticles in the Au–Ge system [12] to stress the importance of system geometry. The bulk thermodynamic and the surface properties given in [12] are used. The calculated liquidus lines are given in Fig. 6. The depression of the liquidus line is more important for the nanowire/nanoparticle geometry than for a system of spherical nanoparticles. Quantitatively, the eutectic point is located at a temperature of 566 K and a composition of 25.5 at.% Ge for a nanoparticle system of 10 nm whereas the nanowire/nanoparticle system has a transition temperature of
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dependence of this interfacial energy are available to the best of our knowledge, we arbitrarily selected a constant value. Consequently, only the surface energy of the solid semiconductor and the surface energy of the liquid alloy will have an influence on the position of the liquidus line. The transition temperature between the initial and the final state is calculated by finding the zeros of the following function, the difference between the Gibbs free energy in the final state and the Gibbs free energy of the initial state: system
1G = Gf
,equi (xliq , T ) − Gsystem (T ) i Au
liq,equi H ,SER = nequi xAu (Gliq ) liq Au − GAu
,equi liq + xliq (GGe − GHGe,SER ) Ge
Fig. 6. Comparison of calculated liquidus lines of the Ge-rich side of the Au–Ge system for individual nanoparticles and a nanoparticle in contact with a nanowire for different wire radii.
520 K and a minimum Ge solubility of 24 at.% Ge. The liquidus temperature at 50 at.% Ge is equal to 869 K in the case of the nanoparticle and 844 K in the case of the system constituted of a nanoparticle and a nanowire. This difference can be explained by the different surface-to-volume ratio of the two geometries. A sphere has namely a surface-to-volume ratio of 3/r whereas the nanowire and nanoparticle structure has a surface-to-volume ratio equal to 2/Rwire + 3(4R2part − Rpart h)/(h2 (3Rpart − h)). For a given particle size, the second expression is bigger than the first one. 4.3. Influence of model parameters The study of the influence of specific surface properties on the phase diagram of small systems can indicate which are critical for the understanding of nano-systems. Surface properties of pure metallic and semiconducting liquids are well known [27]. Surface properties of solids are more challenging to measure and therefore more difficult to find in the literature. For some elements, calculated values based on atomistic modelling can be found. First, we will show analytically how the various parameters impact the results and secondly, we will quantify these effects for different values of the surface properties available in the literature. To calculate the liquidus line, the Gibbs free energy of the final state is minimized, which is equivalent to looking for the zeros of the derivative with respect to the liquid composition of the Gibbs free energy. Since the Gibbs free energy of the final state equals system
Gf
liq liq H ,SER H ,SER ) ) + xliq = nliq xliq Ge (GGe − GGe Au (GAu − GAu liq liq liq + RT (xliq Au ln xAu + xGe ln xGe ) ! X liq liq liq liq v Lv (xAu − xGe ) + xAu xGe
v
+ (4π
R2part
liq − 2π Rpart h)γAu ,Ge
liq,sol sol + π R2wire γAu ,Ge + 2π Rwire lw ire γGe Au,Ge R2wire liq,sol
(30)
the term π γ will not appear in the derivative, as it is composition independent. Since the surface energy of the liquid solution is composition dependent, it would have been logical to use a composition-dependent liquid–solid interfacial energy. However, since no experimental value is available for this interfacial energy and since no model describing a composition
,equi liq,equi ,equi liq,equi + RT (xliq ln xAu + xliq ln xGe ) Au Ge ! X liq,equi liq,equi liq,equi liq,equi v + xAu xGe Lv (xAu − xGe ) v
liq liq,sol 2 + (4π R2part − 2π Rpart h)γAu ,Ge + π Rw ire γAu,Ge sol sol + 2π Rwire lwire γGe − (4π (R0part )2 − 2π R0part h0 )γAu sol,sol 0 sol − π R2wire γAu ,Ge − 2π Rwire lw ire γGe equi
liq,equi
liq,equi
(31)
where nliq , xAu and xGe are the equilibrium number of moles of liquid and molar fractions of the final state. Consequently, it follows from this expression that all surface or interface parameters affect the transition temperature. Furthermore, one can show that there is a linear relationship between the transition temperature and the solid/solid particle/wire interfacial energy and the solid/liquid particle/wire interfacial energy for the Au–Ge system. Since the interaction parameters in the Au–Ge system have a linear dependence with temperature, setting 1G = 0 will lead to a linear dependence between the transition temperature and the solid/solid interfacial energy between the wire and the particle, if all the other parameters are kept constant. We start our study with the influence of the surface energy of the nanowire. In [26], values of the Ge surface energy are given depending on the orientation and on the calculation or experimental method. We selected to work with the (111) orientation, which is the lowest energy orientation, and surface energy values of 1.01 and 1.32 J/m2 . For a wire radius of 10 nm, the calculated liquidus lines are given in Fig. 7. For the 1.01 J/m2 surface energy density, a value of 529 K and a minimum solubility of 23.2 at.% Ge slightly above the transition temperature are calculated. For the 1.32 J/m2 value, we obtain the values quoted above, a temperature of 523 K and minimum solubility of 24 at.% Ge. For higher Ge contents, the difference between the two liquidus lines becomes more important since the surface energy of Ge becomes more and more important in the calculations. In the case of the In–Si system, the same general comments as for the Au–Ge system can be done. The liquidus lines are given in Fig. 8 and have been calculated for a wire radius of 10 nm with a silicon surface energy value of 1.5 J/m2 and a value of 2 J/m2 [26]. A small difference of 2 K is calculated for the transition temperature of the system. We obtain 322 K with a surface energy of 1.5 J/m2 and 320 K with a surface energy of 2 J/m2 . Again, the difference is more important when looking at liquidus temperatures for higher Si contents. The influence of the surface energy of the metal is investigated. As for solid Ge, we performed the calculation of the liquidus line for two values of the surface energy of solid gold, taken from two different sources. The calculations have been performed for a wire size of 10 nm and for gold surface energy values of 1.41 and
Y. Eichhammer et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 173–182
Fig. 7. Influence of solid Ge surface energy on the calculated liquidus line of the Ge-rich side of the Au–Ge system for a wire radius of 10 nm.
181
Fig. 9. Influence of solid Au surface energy on the calculated liquidus line of the Ge-rich side of the Au–Ge system for a wire radius of 10 nm.
Fig. 8. Influence of solid Si surface energy on the calculated liquidus line of the Si-rich side of the In–Si system for a wire radius of 10 nm.
Fig. 10. Influence of liquid Ge surface energy on the calculated liquidus line of the Ge-rich side of the Au–Ge system for a wire radius of 10 nm.
1.25 J/m2 [29]. These calculated liquidus lines are given in Fig. 9. In the first case, a transition temperature of 523 K and a minimum solubility of 24 at.% Ge had been calculated. In the second case, a transition temperature of 540 K and a minimum solubility of 24.5 at.% Ge have been calculated. We note that this parameter has a rather important effect on the value of the transition temperature of the system, as a difference of 0.016 J/m2 in this value leads to a difference of 17 K in the value of the transition temperature. Referring to the expression of 1G, it can qualitatively be explained that the transition temperature is mainly determined by the difference between the liquid surface energy of the nanoparticle and the solid surface energy of the nanoparticle. A study of the effect of the surface energies of the pure liquids gave the following results. No visible difference could be found on the two calculated liquidus lines using the value of the surface energy of liquid gold given in Table 2 and the value of 1.13 − 0.14e − 03(T − TmAu ) given in Ref. [30]. First, these two values are comparable and do not introduce significant differences in the calculation. Secondly, Butler’s equation showed that Ge will have a tendency to segregate to the surface in the Au–Ge liquid alloy (see Fig. 2b). Therefore, the surface energy of liquid Ge will have a stronger influence on the surface energy of the melt than the surface energy of pure liquid gold. In Fig. 10, a clear difference is indeed seen between the calculated liquidus lines using the value
of Table 2 and a value of 0.583 − 0.08e − 03(T − TmGe ) of Ref. [31] for the surface energy of liquid germanium. With the first value, we calculate a transition temperature of 523 K and a minimum solubility of 24 at.% Ge, whereas with the second value we calculate a transition temperature of 501 K and a minimum solubility of 22.5 at.% Ge. Finally, we studied the influence of the solid–solid interfacial energy on the calculations. This value was not found in the literature and the values were consequently chosen arbitrarily. The liquidus lines are given in Fig. 11 for the Au–Ge system. We have a decrease of transition temperature from 546 to 523 K when increasing the value of the solid–solid interfacial energy from 0.5 to 1 J/m2 in a 10 nm system. Similarly, the temperature decreases from 523 to 478 K when increasing this value from 1 to 2 J/m2 in a 10 nm system. The minimum solubility is modified as well. It decreases from 25 at.% Ge to 23.5 at.% Ge when increasing the interfacial energy from 0.5 to 1 J/m2 and it decreases from 23.5 at.% Ge to 20.5 at.% Ge when increasing this value from 1 to 2 J/m2 . In the case of the In–Si system, the liquidus lines are plotted in Fig. 12 and have been calculated for a wire radius of 10 nm. The liquidus lines have been calculated for three different values of the solid/solid interfacial wire/particle energy, 0.5, 1 and 2 J/m2 . We respectively calculated transition temperatures of 358, 322 K and an undetermined transition temperature, due to
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individual spherical nanoparticles to evidence the importance of geometry in the phase equilibria of nano-systems. Appendix. Supplementary data Supplementary material related to this article can be found online at doi:10.1016/j.calphad.2011.02.002. References
Fig. 11. Influence of solid–solid nanoparticle–nanowire interfacial energy on the calculated liquidus line of the Ge-rich side of the Au–Ge system for a wire radius of 10 nm.
Fig. 12. Influence of solid–solid nanoparticle–nanowire interfacial energy on the calculated liquidus line of the Si-rich side of the In–Si system for a wire radius of 10 nm.
the minimum temperature for the validity of Gibbs free energy expressions of 25 °C. For both systems, this interfacial property has an important effect on the calculated transition temperature. For accurate calculations, it is thus important to develop methods to measure or calculate this interfacial energy. 5. Conclusion We have developed a model to describe phase equilibria in a system constituted of a nanoparticle in contact with a nanowire, accounting for the influence of surfaces and interfaces. Equilibrium conditions are obtained by minimization of the total Gibbs free energy of the system. We show that surface properties of the solid semiconductor and liquids influence both the transition temperature between a solid and a liquid nanoparticle and the liquidus temperatures, whereas surface properties of the solid metal and particle/wire interfacial properties only influence the transition temperature between the solid and the liquid nanoparticle. The accuracy of the calculated phase diagram depends largely on the reliability of these properties. Furthermore, the calculated phase diagrams were compared to the case of
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