Gupta D. (Ed.). Diffusion Processes in Advanced Technological Materials

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292 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

plotted as in Fig. 6.3 (namely L

versus t), and extrapolate to some

finite value of t for L  zero, do not in fact satisfy the conditions for

such a plot. Instead, if we want to extract proper values for long-range

diffusion constants, they should be plotted as L versus t

12

, because

extrapolation gives a positive value of L for t  0. The physical cause

of this apparent paradox is that, initially, long-range oxidation is pre-

ceded by a period of anomalously fast oxidation, the source of which is

usually not analyzed in detail.

[12]

There is little doubt, however, that this

complexity arises as follows: The usual analysis of phase formation

kinetics, as above, is based on the assumption that the reactants remain

in their equilibrium state. However, this assumption is invalid during

the initial stage of oxidation, which is driven by very high free-energy

changes. When the oxide becomes thick enough and the rates slow

enough, the reacting surfaces (interfaces) return to equilibrium.

Initially, however, the rate of oxidation is too high to allow the sub-

strate atoms to relax to equilibrium positions and states before being

consumed by the growing oxide. This results in supersaturation of

defects, mostly vacancies during metal oxidation, and vacancies as well

as interstitials during Si oxidation, that indeed manifest their presence

in different ways, for example, dislocation loops, inside the substrate

itself. During that period, the equilibrium values of ∆G are not correct.

Excess values lead to enhanced growth rates and to the necessity of

plotting the data as L versus t

12

6.2.4 Linear-Parabolic Kinetics: Sequential

Phase Growth, Grain Boundary

Versus Lattice Diffusion

If several phases are likely to grow, then Eqs. (12a) and (12b) must be

written as:

dt  a[L

 (aK

)]  b[L

 (bK

)] (16a)

and

dt  d[L

 (dK

)]  g[L

 (gK

)]. (16b)

If phase 1 forms first, the growth of phase 2, from Eq. (16b), requires that:

d[L

 (dK

)]  g[L

 (gK

)], (17)

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REACTIVE PHASE FORMATION, D’HEURLE ET AL. 293

which, for L

 0, yields:

 d  (1K

 1K

). (18)

Phase 2 cannot grow until the first phase has reached a critical thickness L

which increases as d (proportional to D

 ∆G

) increases, and decreases as

the initial rate of growth of phase 2, K

, increases. (If that rate is infinite, as

in the case of pure diffusion control, then L

c  0.) It becomes zero already

if K

 K

; namely, there is no critical thickness for second-phase growth if

the maximum growth rates of the two phases are equal, which is indeed as

anticipated. An increase in K

, like an increase in d, increases L

The literature on silicide formation relating to the reactions of thin

metallic films deposited either on single-crystal or polycrystalline Si bears

ample witness to the fact that phases do indeed form sequentially.

[15, 16]

With a 200-nm-thick film, growth of the first phase consumes all of the

metal present before a second phase begins to grow, for example, Ni

Si or

Si, then NiSi and PtSi, implying that L

is larger than or of the order

of 300 nm.

[17–19]

As was emphasized above, such evidence is no confirma-

tion of a true reaction rate limitation at the beginning, so we should not

argue too much that no apparent activation energies have been determined

for this initial stage of silicide growth. What is certain is that when plot-

ted as in Fig. 6.3, the data show time intercepts that come closer to the ori-

gin as the temperature increases. Thus, if we compare observations made

on thin films, at relatively low temperatures, with observations on bulk

specimens, usually examined on a larger scale and consequently at con-

siderably higher temperatures, in the latter case, the sequential formation

of phases is usually ignored. This is probably more the result of the mode

of observation than of a true difference in behaviors between thin film and

bulk samples, which indeed obey the same laws except in extremely rare

cases (generally with extremely thin layers, below 50 nm). With thin

films, the interfaces tend to be quite clean. Retardation of phase formation

in bulk samples because of interface impurities, such as oxide with Al or

Ti samples, is not discussed here.

In discussing diffusion, the question of whether we refer to lattice or

grain boundary diffusion has been eluded. That was true also when writ-

ing the corresponding equations, which remain the same except for a

change in the value of D. Thus from the kinetics alone, it is difficult to

decide whether growth is due to one or the other of these two diffusion

mechanisms, since if the grain size of the growing layer remains con-

stant, the time exponent 1/n of the growth law remains 12. This ceases

to be true, however, if grain boundary diffusion is dominant and the

grain size increases during compound growth; then 1/n becomes smaller

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294 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

than 12 (because diffusion slows down as the grain size increases). An

example is found in Sn-Ni reactions (for soldering) where a time expo-

nent smaller than 12 for the growth of Ni

is attributed to grain

growth.

[20]

The exponent reduces to 13 if the grain size is proportional

to the thickness of the layer, and even to 14 for normal grain growth,

where the driving force for grain growth is inversely proportional to the

grain size.

6.2.5 First Phase Formed

Alot of ill-spent ink has been poured on the question of what is the first

phase to form. Although it would be prudent to avoid the question altogether

here, it is nearly impossible to do so. According to what has been written in

Sec. 6.2.4, when a new phase is formed from the reaction between two ele-

ments, it is generally true that the free energies of formation of the different

phases do not differ much from each other: factors of 2, 3, or perhaps 4, but

rarely more than that. Thus the competition between phases as to which

would grow first depends on the product D

 ∆G

. It would be dominated

by D

’s, which differ much more widely than ∆G

, so that the first phase to

grow would be the one with the highest D,

[21]

or to be more precise, the

highest value for D  ∆G. An extension of that statement based on some

observations about the relative diffusion of the two atomic species in binary

compounds, somewhat systematized as the “ordered Cu

Au rule,” predicts

that when there is a strong difference in melting points between the reac-

tants A and B, the first phase to form would be a phase rich in the element

with the low melting point, probably the one richest in that element.

[22]

more sophisticated approach,

[23]

based on irreversible thermodynamics,

implies that the first phase to grow should be the one with the highest rate

of free energy dissipation, namely, D  ∆G

. However, inasmuch as the

∆G’s vary little from phase to phase in the initial reaction, the phase with

the highest D  ∆G

product is likely to be that with the highest D.

Regarding the scale at which observations are made, the observations above

would seem to be valid for layers about 20 nm thick. At a smaller scale, we

might need to look into nucleation phenomena, about which little has been

said until now. With silicides, one can examine the process of formation of

amorphous phases reviewed by Sinclair.

[24]

6.2.6 Nucleation of the First or Second Phases

In Sec. 6.2.6, the description of phenomena that occur during what

can loosely be called the incubation time has remained deliberately vague,

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REACTIVE PHASE FORMATION, D’HEURLE ET AL. 295

except that the overall kinetics may be described as proportional to t

, with

n usually chosen for the sake of simplicity as 1, regardless of what is phys-

ically happening during this period. Considerable progress was made with

the discovery that the actual phase growth is preceded by an earlier

process related to nucleation;

[25]

the key to experimental success here was

the use of multilayered films that magnify the initial stage of a reaction in

comparison with the second stage, which is growth. During heating at a

constant heating rate, the evolution of heat occurs as shown in Fig. 6.4.

The first peak at low temperature results from the nucleation of the new

phase (in the case of silicides, often an amorphous one), which nucleates

first, then spreads along the original interface while maintaining a small

constant thickness. It is to be noted that the magnitude of the first exother-

mic peak in Fig. 6.4 is too large to be due entirely to nucleation, which

Figure 6.4 Heat evolution as a function of time during heating at a constant

rate of a reacting diffusion couple in a differential scanning calorimeter. There

are usually two peaks. The first, at low temperature, corresponds to a nucle-

ation step and often a lateral growth of the compound along the interface. The

second peak corresponds to what is usually considered growth itself. Very

interesting results have been obtained using multilayers that allow the relative

magnitudes of the two peaks to be affected almost at will. Does the picture

thus obtained provide a balanced view of what happens in an ordinary simple

diffusion couple?

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296 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

contributes only a fraction of the total heat evolved in this first stage. Such

thermal behavior is not limited to silicides; it is also encountered with

other systems, such as Al-Co.

[26]

Although the literature should be studied

on a case-by-case basis, it is probably correct to say that we still need a

clear analysis of the transition between what occurs in stage 1 and the nor-

mal phase growth in phase 2, as described by Eqs. (4), (8), (12a), and

(12b).

Considerable work has been published regarding the nucleation that

occurs during peak 1 (Fig. 6.4). This research suffers from a number of

difficulties, many of which are common to all studies of nucleation phe-

nomena. The activation energy for nucleation ∆G* is the sum of two

terms:

∆G*  ∆G*

 ∆G*

kin

, (19)

where ∆G*

kin

is the activation energy for the growth of a nucleus of criti-

cal size. In some cases, it may be equal or close to the activation energy

for growth that occurs later, but conceptually it is different. The other term

∆G*

has the form:

∆G*

→∆s

∆G

, (20)

where ∆s is the increase in interface energy on the formation of a nucleus

of critical size, and ∆G is the decrease in free energy (which should be

expressed here per unit volume) on the formation of the new phase. The

signs of ∆s and ∆G are opposite, and for small nuclei, the first term is

dominant. Thermodynamics dictate that the equilibrium density of nuclei

of critical size should be proportional to exp(∆G*

kT). The rate of

nucleation will necessarily be proportional to that quantity, yet nucleation

does not occur unless the critical nuclei grow. Therefore, as written in Eq.

(19), the activation energy for nucleation is the sum of two terms.

Diffusion theory states that the activation energy for diffusion via a

vacancy mechanism is also the sum of two terms: the energy for the for-

mation of a vacancy, plus the activation energy for the motion of these

vacancies. The case of nucleation considered here very closely parallels

diffusion.

Conceptually, the first term for the activation energy for nucleation

∆G*

is the easiest to manipulate, at least on paper. In principle, we know

what surface and interface energies are, so that we have an idea what ∆s

is. In principle also, we can assume that ∆G [in Eq. (20)] is the equilibrium

value for the formation of a new phase, as written in Eq. (1). Or, going

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REACTIVE PHASE FORMATION, D’HEURLE ET AL. 297

further, we can avail ourselves of more precise calculations of phase dia-

grams (CALPHAD) that will take into account the shape of the free energy

versus composition curves for the product and reactant phases. The situa-

tion for ∆G*

kin

is more ambiguous, because the formation of the nuclei may

occur along several competing paths with several different activation ener-

gies, especially when considering the formation of a compound new phase.

In fact, the conditions for any reliable calculations are often quite desper-

ate. Information about surface energies are usually lacking, and we are

unsure of the extent to which bulk or equilibrium free energies are valid for

submicroscopic particles. Nevertheless, it is quite current (although wrong)

to analyze activation energies for nucleation purely with respect to ∆G*

only, neglecting ∆G*

kin

, or assuming that in considering the rate of nucle-

ation, ∆G*

kin

is simply lost in the form of exp(∆G*

kin

kT) among the pre-

exponential terms in front of exp(∆G*

kT).

There may be some justification for neglecting the kinetic term for

nucleation, for example in the study of the precipitation of Co atoms in a

supersaturated solid solution of Co in Cu. That may occur via homoge-

neous or heterogeneous nucleation, and in comparing the two, we can

assume that ∆G*

kin

remains constant, so that experiments and calcula-

tions can be conducted with focus on ∆s (and therefore on ∆G*

) only.

When ∆G in Eq. (20) is very small, ∆G*

becomes very large, so that

qualitatively no harm is being done if ∆G*

kin

in Eq. (19) is neglected.

[27]

However, when ∆G*

kin

is neglected in attempting to determine which

phase forms first, for example, in the Ni-Al reaction,

[28]

which will be

used presently to illustrate difficulties encountered in the analysis of

competitive nucleation between phases of widely different compositions.

The Ni-Al system is almost ideal, because it is nearly completely sym-

metrical, with NiAl in the middle, Ni

Al and NiAl

on opposite sides of

the diagram, plus two intermediate phases, Ni

and Ni

. The

enthalpies of formation of the two extreme phases Ni

Al and NiAl

are

very close: 153 and 150 kJ/mol, respectively.

[29]

(This is surely

within the limits of accuracy of the measurements.) To demonstrate that

NiAl

should form preferentially to Ni

Al, the difference of enthalpies

between the terminal solid solutions is used: Al is somewhat soluble in

Ni; Ni in Al is not. It follows that the free energy of the Al in Ni solid

solution is somewhat lower than Ni in Al. Therefore, the ∆G (or ∆H) for

the formation of Ni

Al is somewhat smaller than ∆G for the formation of

NiAl

. This leads to a somewhat greater value for ∆G*

, and the pre-

ferred formation of NiAl

. However, considering that in one case the

phase is mostly composed of Ni and in the other mostly of Al, it is likely

that the differences in ∆G*

kin

would be 1 eV or more (because of the large

difference in melting points and consequent large difference in the

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298 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

respective diffusion coefficients). Moreover, the enhanced thermody-

namic stability of a solid solution of Al in Ni, as opposed to that of Ni in

Al, implies that in the first case Al atoms will favor Al-Ni bonds, that is,

surround themselves with Ni atoms. Thus, since Ni

Al has the L1

struc-

ture, each Al atom in solid solution in Ni would constitute a potential

Al unitary cell. It is hard to understand that such a configuration

would not favor the nucleation of Ni

Al, independent of what the math-

ematics of ∆G*

imply.

We must also consider the fact that the nucleation of a new phase is

accompanied by sharp concentration and therefore also free energy gradients

because the very first phenomenon to occur is the superficial saturation of

A with B and B with A.

[30, 31]

Figure 6.5 shows that in such concentration

gradients, nucleation would happen preferentially in the phase, for exam-

ple, A, where the concentration gradient is minimum, because the diffu-

sion coefficient is maximum. Thus the phase to be nucleated would likely

be rich in A. From either Eq. (19) or Fig. 6.5, we conclude that the ques-

tion of the first phase to be nucleated is dominated by kinetic considera-

tions rather than by variations in ∆G*

. Moreover, if a phase with small

diffusion coefficients is nucleated first, it is likely to be immediately con-

sumed by another phase with faster kinetics, which has appropriately been

called vampire phase.

[32]

The discussion is carried out in terms of nucle-

ation,

[32]

but very much the same conclusions were reached earlier by sim-

ple considerations of competitive phase growth.

[33]

Thus it does not seem

that the analysis of nucleation is very helpful in understanding much of

the process of phase formation from reacting solid elements. Whether this

statement is generally valid or not, it is supported by a systematic analy-

sis of the reaction of Ti with Al, where experimental observations could

be understood in terms of competitive phase growth alone.

[34]

As antici-

pated, the first phase formed is rich in the element with the lowest melt-

ing point, Al.

Not much more can be said about nucleation in the present context.

It was emphasized above that whether we consider competitive phase

growth alone or nucleation in a concentration gradient (the usual condi-

tion), we are likely to predict that the first phase to grow will be rich in

the element with the lowest melting point. Further, in the reaction of Al

with Ni, NiAl

and Ni

are rich in Al, so that in Eq. (19), the ∆G*

kin

for these two phases might be quite similar. Then in comparing NiAl

versus Ni

, ∆G*

might play a significant role, quite different from

the role it would have in comparing NiAl

with Ni

Al. But in comparing

differences in ∆G*

, we are obliged to factor in the role of elastic energy

∆H

elas

, which can significantly affect the value of ∆G and about which

extremely little is known. In the absence of firm information about ∆s,

that term also remains somewhat unknown. We are left with experimental

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REACTIVE PHASE FORMATION, D’HEURLE ET AL. 299

Figure 6.5 Schematic representation of the initial step in a reaction between two

elements, A and B. The elements are assumed to have symmetrical thermody-

namic properties, as shown (the same mutual solubilities, c

and c

, at equilibrium,

or c

and c

, with the two elements in contact with each other), and similar com-

pounds (for example, A

B and AB

, not yet nucleated in the graph above), but dif-

ferent diffusion coefficients. (The diffusion coefficient of B in A, D

, is approxi-

mately equal to the coefficient of self-diffusion, D

. Both are much greater than

the corresponding coefficients in the other element, B.) The driving force G [Eq. (20)]

for the nucleation of an A-rich or B-rich compound is proportional to the blackened

areas on each side of the interface. Nucleation would then occur first on the side

of the equilibrium diagram where the element has a high diffusion coefficient. It fol-

lows that nucleation tends to be dominated by kinetic factors similar to those that

control growth. The reason for this is that, according to the Cu

Au rule,

[22]

if D

much greater than D

in general, D

A3B

would be much greater than D

AB3

, and

these latter coefficients control growth.

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300 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

observations, which show that the reaction of Ti with Al produces

metastable forms of TiAl

with structures at variance with the equilib-

rium structure,

[35]

while Ni-Al reactions can also lead to a metastable

phase with a composition not encountered in the equilibrium diagram

[36, 37]

(This appears to be in accordance with the Cu

Au rule

[22]

since, other things being equal, Al should be more mobile in Al

than

in Al

Ni.) Yet it is likely that nucleation processes play a role in deter-

mining which one of these phases is formed. However, conceptual

analysis has been of little help in resolving this problem. Small differ-

ences in sample preparation and heat treatment affect the observed

results, which does not simplify any realistic analysis. The complexity

is increased in a system such as Al-Pd, where many aluminum-rich com-

pounds are stable; reports of the first phase to form tend to change with

the investigators.

[38]

Whatever role nucleation plays in the selection of the formation of

these first or second phases, the relatively large ∆G in Eq. (1) [although

it is not the precise value that should be entered in Eq. (20)] results in a

relatively easy nucleation process and in a great abundance of nucle-

ation sites. Apractical effect of the numerous nucleation sites is that for

layers at least 10 nm thick, the interfaces are planar and parallel, as in Figs.

6.2 and 6.3. This is particularly noticeable with silicide reactions, where

the Si substrates are perfectly specular to begin with, and the samples

remain equally specular after the deposition of some 100 nm of Ni, as

well as after annealing to form Ni

Si and then NiSi. There are two gen-

eral aspects to be considered: (1) Reactive phase formation is usually

dominated by diffusion considerations, rather than nucleation. (2) In the

same way, kinetic factors, notably the growth modulated by diffusion,

dominate over thermodynamic ones. The fact that the first phase to grow

does not seem to depend significantly on the state of the reactants (solid,

liquid, or gas) provides experimental support for the first point. For

example, the disilicides NbSi

and MoSi

are obtained from the reaction

of thin metal films with Si substrates.

[39]

However, the same products are

obtained from the reaction of the metal with molten salt

[40]

or with SiH

(a gas),

[41]

respectively. The compound Cu

is obtained indifferently

whether Cu reacts with molten or solid Sn.

[20, 42, 43]

Although thermody-

namics is not unimportant, it indicates mostly what cannot happen: A

system cannot evolve in such a way as to increase the overall free energy

level. In the reaction of Mo with SiH

, MoSi

cannot be formed if the

pressure of SiH

and, consequently, the activity of Si are too low.

Gagliano et al.

[43]

provide a very good example of an experimental study

of nucleation in the reaction of Cu with liquid Sn. The potential role of

nucleation at the start of a reaction was theoretically investigated by

Gusak et al.

[44]

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REACTIVE PHASE FORMATION, D’HEURLE ET AL. 301

6.2.7 Nucleation-Controlled Reactions and

Consequences: Sequence of Phase

Formation, Bulk Samples, Stresses

If we follow the example of the formation of Ni silicides, we find

that with excess Si, the equilibrium phase should be NiSi

. However,

the formation of the equilibrium phase does not occur smoothly after

the formations of the two previous phases. These processes were dif-

fusion-controlled and occurred at about 350 and 400°C, respectively.

But once NiSi is formed, further heating does not cause any significant

change until a temperature above 750°C is reached. Then suddenly,

and apparently explosively, the sample transforms to NiSi

, with the

sample assuming a new frosted-glass appearance that is quite visible

to the naked eye. The nucleation of the new phase in this case was so

difficult that it required a relatively high temperature, where for sam-

ples of the order of 400 nm thick, diffusion was no longer rate-limit-

ing. The reason is simple: The overall ∆G (∆H) for the formation of

Si from Ni and Si is 143 ± 11 kJ/mol; that for the formation of

NiSi from the reaction of Ni

Si with Si is 18 ± 9 kJ/mol;

[27]

but for

the reaction:

NiSi  Si  NiSi

, (21)

the ∆H (∆G) is only 5 ± 20 kJ/mol. It is so small that it is not easily

accessible from experimental measurements. When the ∆H’s and ∆G’s

become small, it is no longer correct to consider that the two sets of val-

ues are identical.

[45]

However, what matters presently is that both values

are indeed quite small, so that from Eq. (20), ∆G*

becomes very large.

The practical consequences of the high temperature required for nucle-

ation to overcome this problem, the paucity of nuclei, and the rough sur-

face of the silicide layers are considered in Sec. 6.3. Even without knowing

the quantitative details of the nucleation process, games can be played:

Replacing monocrystalline Si with amorphous Si does not significantly

change the formations of either Ni

Si or NiSi, but the formation of NiSi

becomes diffusion-controlled.

[46]

The increase in ∆G, for Eq. (21), equal

to the difference in free energy between crystalline and amorphous Si, is

enough to cause a change in the mode of formation of NiSi

. Errors have

been made in interpreting reactions such as the formation of NiSi

, where

the sudden occurrence leads to the conclusion of a reactive explosion.

This is wrong. The nature of the reaction is dominated by a small ∆G and

∆H, and cannot therefore be truly explosive, especially on Si substrates

with a high coefficient of thermal conductivity, which limits any self-

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