Criterion for silicon formation in transition metal-silicon diffusion couples

Pergamon

0008-4433(94)0002@-4

CRITERION FOR SILICIDE FORMATION IN TRANSITION METAL-SILICON DIFFUSION COUPLES

LIN ZHANG and DOUGLAS G. IVEY Department of Mining, Metallurgical and Petroleum Engineering, University of Alberta,

Edmonton, Alberta T6G 2G6, Canada

Abstract-In this paper, a criterion for silicide formation in metal-silicon diffusion couples, based on the rate of change of free energy, referred to here as the free energy degradation rate (FEDR), has been developed from a kinetic model for silicide formation. In the kinetic model, silicide formation is divided into three steps: diffusion of the predominant diffuser (or moving reactant) to the reactive interface. followed by release of the less mobile species (non-moving reactant) from its lattice and intermixing with the moving reactant, and finally formation and growth of the silicide phase. It has been shown that the free energy change due to silicide formation in a diffusion couple can be determined by examining the free energy change of the reaction region (or reactive interface) located between the growing silicide and the non-moving reactant phase. The free energy degradation rate per unit area of a given reaction region can be expressed as a sum of three contributions. each corresponding to one of the three steps. Each term is a product of a thermodynamic flux and a driving force. These fluxes and driving forces have been examined individually; by analyzing how they change with time, it is shown that when a number of possible reactions compete with one another in a reaction region. there always exists a reaction that will result in the largest FEDR in this region. It is also shown that the largest FEDR leads the system to a relative minimum free energy state that is most stable compared with any other energy state at a given instant.

Based on these results, a criterion for silicide reactions has been proposed. During silicide reaction in a reaction region of a metalHi diffusion couple, there are always a number of possible reactions competing with one another. The reactions which result in the largest FEDR will actually occur. This criterion combined with the new kinetic model has been successfully applied to predict silicide formation in 15 metal-Si systems.

R&urn&Dam cet article, on a developpt un crittre pour la formation de siliciures dans les couples de metal-silicium fondt sur la vitesse de changement d’energie libre, appelee ici vitesse de degradation de I’inergie libre (FEDR); a partir d’un modele cinetique de formation des siliciures. Dam ce modele cinetique. la formation des siliciures est divisee en trois &tapes: la diffusion du diffuseur principal (ou reactif mobile) vers les interfaces reactives. suivi par la liberation des especes les moins mobiles (rtactifs immobiles) dans son reseau qui se melangent au reactif mobile, puis tinalement par la formation et la croissance de la phase de siliciure. On a montre que l’on pouvait determiner le changement en energie libre dd Q la formation de siliciure dans un couple de diffusion en examinant le changement en energie libre dans la region des reactions (ou interface reactive) sit&e entre le siliciure croissant et les phases reactives immobiles. On peut exprimer la vitesse de degradation d’energie libre par unite de surface pour une zone de reactions don&e comme &ant la somme de trois contributions correspondant a chacune des trois &apes. Chaque terme est le produit d’un flux thermodynamique et d‘une force potentielle de transformation. On a examine ces flux et ces forces thermodynamiques individuellement. Elles changent avec le temps. Ainsi nous avons montre que lorsqu’un nombre de reactions possibles dans une region, entrent en concurrence les unes avec les autres, il existe toujours dans cette region, une reaction pour laquelle la FEDR est la plus grande. Nous avons aussi montre que la plus grande FEDR conduit le systeme a un (itat d’energie libre minimal relatif qul est le plus stable par comparaison avec tout autre &at d’energie a un instant dorm&

NOUS avons propose un crittre base sur nos rtsultats. pour les reactions des siliciures. Au tours des reaction des siliciures dans la zone de reaction d’un couple de diffusion metal-Si. il y a toujours un certain nombre de reactions possibles en concurrence les unes avec les autres. Les reactions qui rtsultent de la plus grdnde FEDR se produiront effectivement. Nous avons applique avec succes ce critere, combine au nouveau modele cinetique, pour predire la formation de siliciures dans 15 systemes metal-Si.

INTRODUCTION to as an empirical rule for predicting first-phase nucleation in

thin film metalHi couples. According to this rule, “the first Prediction of silicide formation sequences is a long-standing compound nucleated in planar binary (thin film) reaction problem for kinetic studies. A number of empirical rules and couples is the most stable congruently melting compound adjac- theoretical models have been proposed over the years [l-21]. ent to the lowest temperature eutectic on the bulk equilibrium

Before solid-state amorphization reactions were found to be a phase diagram.” A similar rule was also proposed for first phase rather general experimental phenomenon in thin film diffusion nucleation in metal-metal thin film systems [22]. couples, the Walser-Bent: rule [I] had been frequently referred A similar type of model was formulated by Tsaur et ul.

51

52 L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL--Si DIFFUSION COUPLES

[2] after examining phase diagrams and sequences of phase formation in several metal-silicon systems. “The second phase formed is the compound with the smallest AT that exists in the phase diagram between the composition of the first phase and the unreacted element.” They defined AT as the temperature difference between the liquidus curve and the peritectic (or peritectoid) point for the system under consideration. This rule was successful in explaining the formation sequences in Pd-Si, Pt-Si, Ni-Si. Co-Si, and some other systems.

A more recent model, i.e. the ‘effective heat of formation’ model, has been proposed by Pretorius to predict single-phase growth sequences in metal-Si and metal-metal thin film couples [14-l& 23-251. In this model, the effective heat of formation, AH’. is defined by the expression [l&16]:

AH’=APx !

available concentration limiting element compound concentration limiting element ’

(1)

where AH” is the standard heat of formation of the silicide in joules per mole of atoms. When calculating AH’ for a given metal-Si binary system, the composition at the lowest eutectic point is taken as the available concentration limiting element, which is based on the argument that the intermixing at the metal-Si interface, during annealing, takes place at con- centrations similar to that of the lowest eutectic point. In addition. another condition is imposed, i.e. that nucleation of a congruent phase is much easier than that of a noncongruent phase. Using this condition and the term AH’, an “effective heat of formation” rule for single phase growth sequences has been formulated by Pretorius [14, 151: “The first silicide compound to form during metal-Si interaction is the congruent phase with the most negative effective heat of formation (AH’) at the concentration of the lowest temperature eutectic of the binary system. The next phase formed at the interface between the compound and remaining element (Si or metal) is the next congruent phase richer in the unreacted element, which has the most negative effective heat of formation.”

The three models briefly discussed above are very similar in their approach to silicide formation. The main difference between these approaches is that thermodynamic data, AH, instead of equilibrium phase diagram information, are used in the effective heat of formation model. I f the experimental results from early studies are used, predictions of first phase formation from Pretorius’ model are about 80% successful [1416, 23- 25].

There are several problems with the three models given above and these are addressed below. Both the Walser-Ben& rule and the effective heat of formation rule imply that there is only one initial phase in a given metal-Si diffusion couple, whereas more recent experiments provide evidence of different first-phase formation being possible. First-phase formation predictions from these two rules and experimental results from I5 metal-Si systems are shown for comparison in Table 1.

A second problem with the above models revolves around solid-state amorphization and metastable phase formation. From the three rules introduced above, all the phases formed should be equilibrium phases present in the phase diagrams. During the last few years, a number of research groups have

reported that amorphous layers formed initially and grew before any crystalline phase formed. when a thin film metal-Si diffusion couple was annealed at relatively low temperatures [39, 77, 78. 98-l 141. The metal-Si systems which have shown such kinetic behavior include Rh-Si [99], Ti-Si [39, 100-l lo], Ni-Si [I 11. 1121: Cr-Si [98], V-Si [39], Zr-Si and Hf-Si [77, 78, 981, and Nb-Si and Ta-Si [I 13: 1141. In these systems, the amorphous phase is the initial phase when a given system is annealed at low temperatures. Other metastable phases that can form during annealing of thin film couples have also been reported. A well known example is C49 TiSi?, whose formation precedes that of the equilibrium phase C54 TiS& [27, 28, 100, 10 I, 103, 110, 115-l 191. Similarly, in the Mo-Si system, growth of hexagonal MoSi, occured before the formation of tetragonal MoSi, [120]. Some other examples are metastable &Ni$i [I211 and metastable hexagonal Nisi [68, 1221 formation on Si(1 I I), respectively, and metastable FeSi, phase formation on Si( 111) [ 1231. Finally, when the Walser-BenC rule and effective heat of formation rule were proposed, it was pointed out by the authors that there existed some exceptions to their rules; e.g. Zr,Si, [I] and Mn,Si, [I, 151 were predicted from the rule as the initial phases in Zr-Si and Mn-Si systems. Mn,Si [43, 441 or MnSi [45, 461, and ZrSiz [Z, 29, 38, 50. 51, 75, 761 or ZrSi [77, 781 have been reported to be the initial phases in the Mn-Si and Zr-Si systems, respectively.

A more advanced model linking thermodynamics to kinetics has been recently proposed by Be& [9]. After reviewing the work being done on other crystal growth phenomena. such as dendrite nucleation and growth, and snowflake generation, Benk pointed out that these systems change their shape as a function of size because the new metastable shape allows a more rapid decrease in the system energy. It was argued that the metastable shape of these systems can be analogous to the metastable chemical structure that forms in metal-Si couples [9]. As a result, Ben& proposed a criterion for silicide nucleation, i.e. the maximum free energy degredation rate (FEDR) or the largest FEDR. According to the criterion, at the beginning of the reaction the phase whose formation results in the maximum FEDR grows first. When the growth of a new phase reduces the energy of the system faster than continued growth of the first phase, the new phase will start to nucleate [9]. This criterion appears to be logical, since the largest FEDR implies that under given kinetic conditions the free energy of the reactive interface and surrounding region is a minimum relative to all energy states accessible by other reactions. The state determined by the maximum FEDR criterion is stable in terms of minimum free energy at given kinetic conditions and time.

The term FEDR is expressed by -DAG [9], where D is the interdiffusivity of the moving species in a given silicide, and AC is the free energy change for this phase formation. This model and the maximum FEDR criterion have been used to explain amorphous phase formation at the beginning of silicide reactions [9]. Although an amorphous phase has a smaller value of -AC than stable silicide phases, it can still form initially if its D is large enough that its - DAG is larger than those for other phases. From approximate free energy calculations, Ben& [9] also concluded that growth of amorphous phases is possible between Si and Ni, Co. Pd, Pt, Ti, V, Ta, and Hf, but not between Si and Cr, MO, and W. In contrast, amorphous phase formation at the beginning of silicide reactions has been found

L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL-Si DIFFUSION COUPLES 53

Table I. Comparison of first place predictions with experimental results from the literature

First phases First phases predicted from predicted from W-B rule” [l] EHF rule’ [I51

-_I_-_~-__ TiSiz TiSi:

First phases observed in experiments

--__ TiSiz TiSi

VSi,

CrSi, Mn&

FeSi CozSi

Ni,Si

Zr,Si,

NbSi: MoSi,

VSi,

CrSi, MnSSi3

FeSi Co$i

Ni,Si

ZrSiz

NbSiz MoSi>

Ti,Si, VSiz V,Si CrSi2 Mn,Si MnSi FeSi Co$i CoSi CoSi, CoSi, Ni,Si Nisi Nisi> ZrSiz ZrSi NbSiz MoSi, MoSSi, and MO,%

Pd,Si HtSi TaSi, WSi2 Pt:Si

PdzSi

TaSi, WSi, PtzSi

Pd,Si HfSi TaSiz WSiz Pt$i PtSi

j( W-B rule-Waker-Be& rule. A EHF rule-effective heat of formation rule.

experimentally in Ni-, Ti-, V-, Ta-, Hf- and Cr-Si systems [39,77,78,98-l 141 but not in Co-, Pd-, Pt-and MoSi systems [32, 79, 103, 1061.

Be&s criterion has been used by other groups as well [12, 581. Giisele and Tu [12] have applied the concept of FEDR to amorphous phase formation. At the beginning of the reaction in a metal-% couple, the FEDR for amorphous phase formation is the largest, so it grows first. The FEDR will decrease with phase growth. As soon as the FEDR becomes smaller than that for the crystalline phase to form, the amorphous phase starts to shrink and the crystalline phase grows. The thickness of the amorphous phase at the point where the FEDR for the amorphous phase and that for crystalline phase are equal is the critical thickness, ~7’. These theoretical results have been used to qualitatively interpret the experimental phenomena of amorphous phase formation and subsequent crystalline phase formation in multilayer samples [ 121.

There are some problems with Bene’s criterion. First, the criterion (and the expression) is only qualitative. so that it is not applicable to actual prediction of formation sequences. Secondly, the expression for FEDR. i.e. -DAG, is only applicable to diffusion-controlled growth processes, not to nucleation controlled and interfacial reaction controlled processes. First-phase formation may be a competition between nucleation of all possible phases. After this, the competition is between the growth of the existing phase and the nucleation of other possible phases. As a result. this expression cannot practically be used to determine formation sequence.

From the discussion above it can be seen that the concept

References

[2, 8, 26-281 [29-331

~291 [2, 8, 34-371

[37, 381 [2, 8, 26. 39-421

]43,441 [45. 461

[2. 8, 47-491 [2, 8,29, 38, 5@543

[38, 39. 55. 561 [57. 581 [59-621

[2, 8, 63-691 [7G72] [73. 741

[l, 2, 29, 38, 50, 51, 75, 761 [77. 781 [I. 81

[I. 8, 26, 79-811 F21

[7. 8, 26, 29, 38, 50, 51, 83-851 [8. 86. 871 [S. 88. 891

[I, 8, 29, 511 [7, 8, 29, 38, 50, 51, 84, 9&95]

[96. 971 __I-- .-____~

that the free energy changes as a diffusion reaction proceeds, has been used, more or less. by many researchers [9, 12214, 16, 22-25, 58, 98, 1251. It should be pointed out that since free energy in a diffusion couple must change with annealing time, the free energy change or the FEDR can also rellect the kinetic characteristics of the changing system. Therefore, the study of FEDR and its role in determining formation sequences is definitely significant to the kinetic studies of silicide formation. In this paper, we examine the concept of FEDR and attempt to derive relationships to predict phase formation sequence in metal-silicon diffusion couples, addressing the subjects of first phase formation, metastable phase formation and formation sequence. Many of the ideas expressed below follow from a kinetic model recently developed by the authors [126-1291. In this model, only reactions in which one of the two reactants diffuses much faster in the growing silicide are considered (referred to as Type II reactions). The predominant diffuser is considered to be the moving reactant (M) and the other reactant is the non-moving reactant (N). This type of reaction is quite common in thin-film silicide reactions because these reactions usually occur at temperatures below half of the lowest eutectic temperature in the binary phase diagram for a given diffusion couple. In practice, M could be metal atoms (e.g. near-noble metal-Si diffusion couples) or Si atoms (refractory metal-Si diffusion couples). N could be Si atoms (near-noble metal-Si diffusion couples). metal atoms (refractory metal-% diffusion couples), or silicide ‘molecules’.

The basic assumptions of the model [ 126, 127. 1291 are: (1). in a silicide reaction the reactive interface between the growing

54 L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL-Si DIFFUSION COUPLES

phase (product) and the contracting phase (made up of N atoms or molecules) is the reaction region; and (2) the reaction process can be divided into three steps. Step I : M atoms diffuse through the growing phase into the reaction region. Step 2: M atoms in the reaction region interact with the N atoms that are exposed to this region, which causes the N atoms to be released from their lattice into the reaction region and. at the same time, form product ‘molecules’ with the M atoms. The composition and chemical bonding of the product ‘molecules’ are very similar to those of the growing phase. Step 3: the product ‘molecules’ rearrange themselves on the lattice of the growing phase.

Three physical quantities are used to describe each of these steps. They are the diffusion flux (J) of M to the reaction region, the release rate (r) of N and the formation rate of growing silicide (F). The relationship among these physical quantities will control the actual reaction process, which can be demonstrated by a reaction process plot for a given reaction region [126, 1291. A schematic reaction process plot (RPP) for reactions between a moving reactant and a non-moving reactant is shown in Fig. 1. The release rate is plotted on the vertical axis, while the diffusion flux is plotted on the horizontal axis. The relationship between r and J is indicated by the solid curves. Three overlapping curves, corresponding to three different sili- tides. are shown (additional curves would be required for additional silicides). The slope of the inclined portion of each curve represents the composition of the silicide, C, = m,/n,, where m, and n, are the number of moving and nonmoving reactant atoms (or formula units, f.u.) per formula unit of ith silicide (M,N,,). The abrupt changes in slope, from inclined to horizontal, represent upper critical fluxes (J,,,<) and correspond to maximum release rates for each silicide. Formation rates (F) are also expressed by the inclined dashed lines (overlapping the r vs J curves in places) in Fig. 1, in terms of consumption rates of the reactants, i.e. the moving reactant, mF, and the nonmoving reactant, nF. In many cases, a given silicide of composition C, can exist in more than one form or phase, i.e. it can form as a metastable phase(s) (e.g. amorphous phase) or as a stable phase. Each of these will have a maximum formation rate. The maximum formation rates for two possible forms of silicide 2 are shown in Fig. I by crosses-additional forms would require additional crosses. The same approach is applicable to silicides C, and C,.

The use of a reaction process plot can be demonstrated through the following simple example and Fig. 1. At a given diffusion flux, J,, all possible reactions in the reaction region can be found by drawing a vertical line at Jo. The intersection of the ith inclined solid line with the vertical line indicates the release rate, Y, for ith silicide formation. If it is assumed that the silicide producing the largest release rate at Jo is the one that will form (this assumption will be justified in this paper), then silicide 2 should form. The consumption rate is also indicated by this figure. The consumption rate at Jo exceeds the maximum consumption rate for phase (or form) 2 of silicide 2, i.e. nF,,

-4s a result, phase 1 of silicide 2 will grow. As the diffusion flux decreases during growth of phase 1 of silicide 2, the release rate r2 decreases and passes a critical value, nFzz, in Fig. 1. I f phase 1 is the stable form of C2, phase 1 will continue growing. If, however, phase 2 is the stable phase and phase 1 is metastable, phase 2 will start to nucleate and grow at the interface between the non-moving reactant and phase 1. When the diffusion flux

J3uc fzuc JIUC

J and mF -

Fig. 1. Schematic reaction process plot. J,, is a given diffusion flux; J,uc indicates the critical values of diffusion flux; Y: is the largest release rate at a given diffusion flux J,; nzr’,, and n2Fzz represent the maximum

consumption rates for two forms of silicide 2 (C2).

decreases and reaches r3maX, silicide 3 (C,) will start to form at the silicide 2/non-moving reactant interface. By using this plot. the reaction process can be described clearly. It should be emphasized, however, that only a single RPP is necessary for this type of analysis if the N and M are the same during these reactions and the R region under consideration is the one adjacent to the nonmoving reactant. Otherwise, other RPPs should be used. Furthermore, if a quantitative reaction process plot for a transition metal-Si system is available, the silicide formation sequence can be predicted.

One important advantage of this kinetic model is that the reaction process in a reactive interface is divided into successive steps. Accordingly, the silicide that will form is determined by step 2, and the form of this silicide is determined by step 3. From this point of view, one can explain why metastable phases or even amorphous phases are formed. Another advantage is that this model predicts that. theoretically, any silicide in the equilibrium phase diagram of a metal-Si diffusion couple can form first if the release rate for this reaction is on the stepped curve of the reaction process plot (the reasons will be discussed in detail in this paper), and if the initial diffusion flux is between the lower and upper critical fluxes, respectively, for this silicide [126,128]. Therefore, this model can handle the controversial “first phase formation” problem quite well. Furthermore, the model is able to predict when and which new silicide starts to form in a diffusion couple, providing that reaction process plots for this couple are available. This model has been successfully used to explain some complex phenomena encountered in thin silicide film growth processing [126-l 291.

There may be some concern about the terminology of FEDR, since it deals with the subject of thermodynamics of irreversible processes (TIP), as the system of interest is not in thermodynamic equilibrium and a change in state occurs at a finite rate [130]. It appears, at first glance, that a free energy degradation rate may not be a valid concept in TIP. This is because the only general criterion of irreversibility is given by the entropy production (per unit time) according to the expression, dJ/dt > 0 (where d,S/dt is the entropy production per unit time

L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL-5 DIFFUSION COUPLES 55

due to changes within the system) [I 31-l 331. Actually, as is pointed out by Prigogine [131], other criteria of irreversibility, similar to d,S/dl, exist for some particular cases, such as changes in state at constant temperature and volume or at constant temperature and pressure, to which the thermodynamic func- tions Helmoholtz free energy and Gibbs free energy (G) are applicable. Examples using the rate of free energy change dG/dt to discuss chemical reactions in an ideal gas at constant temperature and pressure can also be found in the literature [133]. Therefore, the criterion d,G/dt < 0 is applicable to irreversible processes during which the temperature and pressure are constant. In textbooks on thermodynamics, such as Ref. [ 1301, the term T(d,S/dt) (where T is temperature) is referred to as the rate of energy dissipation. Since r(d,S/dt) and d,G/dt can be used for different systems, d,G/dt should have another name. Therefore, if the term FEDR due to Ben& is used to indicate d,G/dt, the Gibbs free energy decrease per unit time due to changes within the system, it should not cause any confusion. This definition will be used throughout the following analysis.

EXPRESSION FOR FEDR IN A REACTION REGION

Basic equation for the rate cf free emrgy chan,ge in a systm

From TIP theory, entropy has the following properties [131l 1331:

(I) The entropy of the system is an extensive property. I f a system consists of n parts, the total entropy, S, of the system is

s= is,. ,= I

where S, is the entropy of the,jth part. By differentiating equation (2) with respect to time. one obtains

The entropy change per unit time in the system is equal to a sum of the changes in each part.

(2) The entropy change for a system can be split into two parts, i.e.

dS = d,S+d,S, (4)

where d,S denotes the flow of entropy due to interactions with the exterior, and d,S represents the entropy production due to changes within the system. When the rate of entropy change is of interest, it can be expressed as

dS d,S d,S -=- dt dt +dt’ (5)

The entropy production per unit time, d,S/dt provides a general criterion of irreversibility for TIP, i.e. for a reversible process in the system:

For an irreversible process in the system.

The same is true when the jth part of the system [see equations (2) and (3)] is under consideration, i.e.

and

dS, _ d,S, 0, dt -dt+dt

where d,S,:dt is the flow of entropy due to the interactions of the jth part with the other parts and the exterior of the system and d,S,/dt is the entropy production per unit time due to the changes within the jth part.

It is easy to show that Gibbs free energy has similar properties to those of entropy, so that one has the following equations:

3 _ d,G, W, dt -dt+dt

d,G 2 d 0. dt

The last equation is a criterion of irreversibility for a system at constant temperature and pressure.

In TIP, an analytical expression for the so-called local entropy production per unit time, (d,S/dt)/V (entropy production per unit time and unit volume), has been given as [ 1301:

where J,, is the generalized thermodynamic flux. such as heat flux, diffusion flux, and chemical reaction rate, while X, is the generalized thermodynamic force. An expression similar to equation (13) should also exist for (d,Gidt) if the four assumptions [I 321 for deriving the equation are applicable to the system of interest. The assumptions are the following:

(1) The system is isothermal (T = constant). (2) The system is at mechanical equilibrium (no mass flow)

and is not subject to external fields. (3) The concentration gradients are not too high, in the sense

that the composition variables do not vary appreciably within distances of the order of the mean free path. This restriction implies, in particular, the absence of interfaces inside the volume V.

(4) The system is subject to time-independent boundary conditions

The third assumption [ 1321 cannot be satisfied if the reaction (R) region or reactive interface is considered as a part of the system (e.g. a diffusion couple). I f entropy production at the interface region is to be investigated. the contribution of concentration gradient to (d,S/dt) still has to be incorporated into the expression. A generalized expression for such systems is not available at this time (to the knowledge of the authors), so that

56 L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL-% DIFFUSION COUPLES

the expression of d,Gjdt for an R region has to be treated as an individual case.

Also, it should be mentioned that Bene [9] and Giisele and Tu [ 121 have already given the expressions of dG/dt according to their respective models. These expressions are, however, obviously not applicable to the model proposed by this study. This is because the reaction process is considered as a three- step process in the new model and each step has its own contribution to the total d,G/dt in the R region, which cannot be expressed by either Bene’s formula [9] or that of Giisele and Tu [12]. Therefore, an expression for FEDR applicable to this model can only be derived from equations (lo)-(12) and the assumptions of the model.

FEDR expression for a reaction region

In deriving an expression for the FEDR in a reaction region, the following assumptions are made:

(1) Silicide formation in a diffusion couple occurs at constant temperature and pressure.

(2) The diffusion couple is a closed system, i.e. there is only energy exchange between the system and the sur- roundings and there is no matter exchange.

(3) There is no external force or potential field, such as an electrical field, acting on the system. The energy exchange is only due to heat flow.

For a metal-Si system with metal atoms as the moving reactant (the derivation is also applicable to cases where Si atoms are the moving reactant, but it is more convenient to describe the concept when the moving reactant is specified), which is schematically shown in Fig. 2, the free energy of the system at time t can be expressed as

G = G,,+G,+Gs,, (14)

where Gsi, G, and G,, are the free energies of the Si, the reaction region (or the silicide/Si interface) and the region including both the metal and the silicide phase (MS), respectively. From equations (10) and (14), the rate of free energy change in the system is a sum of contributions from the three parts. i.e.

dG dG,, dG, dG,s dt=dr+dt+ dt

The dGus term in equation (I 5) can be split into three parts:

d&s dzGv,s dt dt

+ U&s + 4&s dt dt ’

(16)

where d,G&dt denotes the free energy change rate due to energy exchange between the MS region and the environment

I+ MS region *

Rcactmn region

Fig. 2. A schematic transition metalK diffusion couple divided into three regions, i.e. MS region. R region and Si region. (Not to scale.)

and d,G,s/dt is due to energy and matter exchange betwreen the MS region and the R region, for example metal flux into the R region and silicide molecule flux out of the R region. The last term, d;G,Jdt, is due to the diffusion of metal atoms in the MS region. Therefore, equation (16) has the same form as equation (11) if the first two terms in equation (16) are combined so that the resulting term indicates the contributions due to energy and matter exchanges with neighboring parts of the system and due to energy exchange with the exterior of the system. The three- term expression in equation (16) is preferred because it is more useful in the following discussion. The same is true for equations (17) and (18) below.

The dG,/dt term in equation (15) can be split into four parts,

dG, d,GR d,,GR dczGR d,G, -z- ~ ~ ~ dt dt + dt + dt + dt ’ (17)

where d,G,/dt is due to energy exchange between the R region and the exterior and d,, G,/dt and d,,G,,/dt are due to the matter and energy exchange between the R region and the MS region and the Si, respectively. The term d,G,/dt is due to the three- step process occurring in the R region and will be discussed in detail later. The last term in equation (15) can be expressed as

dGs, d,Gs, d,Gs, d,Gs, __=~ ~ __ dt dt + dt + dt ’

where d,G,,/dt: d,G,,/dt and d,Gs,idr represent the free energy changes per unit time due to energy exchange of Si phase with the exterior, matter and energy exchange between Si and the R region and the diffusion of metal in Si.

It can be shown, by the method used by Prigogine in chapter 2 of Ref. [131], that (d,G,,:‘dr) = - (d,,G,/dr) and (dcZGR/dt) = - (d,G,Jdt). As a result, one obtains, by inserting equations (16))( 18) into equation (15),

dG d,G d,G,s d,G, d,G,, -= dt dt+dt+dt+dt. (19)

The first term in the right-hand side of the equation indicates the total contribution due to energy exchange between the system and the exterior, i.e. it incorporates (d,G,s/dt), (d,G,/dt) and (d,G,,/dt). From equation (12) it is clear that the criteria for an irreversible process to occur in each part of the system are

W,s < o d /G d ,Gs, ~ < 0 and ~ dt ’ dt dt < 0, (20)

which also satisfy the following condition:

d,G diGMs d,Ga d,Gs, < o, -=~ dt dt +t+- dt (21)

According to equations (19)-(21), it is evident that a knowledge of the FEDR due to changes in the R region is sufficient for examining silicide formation which takes place at the reactive interface in a diffusion couple.

In TIP theory, d,S/dt is usually used to describe entropy production at a fixed spatial location. The R region under consideration in this study, however, is moving during silicide formation. Therefore, how d,G,idt for an R region is related to the changes (reactions) in a particular location of the diffusion

L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL-% DIFFUSION COUPLES 51

couple should be explained. In Fig. 3, the diffusion couple shown in Fig. 2 is redrawn with the SI part divided into n slabs, each having the same width as the R region. If the free energy change in each of the slabs is observed by a monitor fixed to the diffusion couple, then the FEDR, d,G,/dt, due to the Ith slab being transformed from Si to silicide, can only occur at time t,, when the R region reaches this slab. Using this approach, the FEDR of the reaction, d,G,jdf, can be examined one slab at a time, which can be quite tedious. An alternative approach is to monitor the R region and observe the FEDR of the R region only. It appears that d,G,/dt is the FEDR of a fixed location (the R region) during the reaction period. Actually, the observed d,G,jdt at instant t, is equal to d,G,/dr due to changes in the Ith slab. In the followmg discussion, the latter approach and the term FEDR of the R region, d,G,/dt will be used in the context described above.

As is assumed in the kinetic model [ 126, 127, 1291 (see Intro- duction), the reaction process in the reactive interface can be divided into three steps. In step 1, M atoms diffuse from one side of the R region (adjacent to growing silicide) to the other side of the region. (In the assumption, step 1 includes diffusion through the growing phase. Here, as is shown above, when only the FEDR for the R region is examined, it is not necessary to consider the diffusion process occurring in other parts of the system.) In step 2. N atoms are released from the surface of their own lattice and form product “molecules” with the M atoms at the same time. In step 3, the “molecules” are rearranged onto the lattice of the growing silicide. The three steps include all possible changes in the R region. Therefore, the FEDR per unit area of R region can be expressed as

ds’R 1 dGR -= dt -/ = J,,A~~+~,AG,~+F,~AG,~, A dt (22)

where dg’,/df is the FEDR for ith silicide formation per unit area of R [region (J/cm2/s). J,, is the diffusion flux of M atoms (atoms&m’/s). (It is expected that the diffusion flux in the R region, J, is equal to the flux into the region from the growing silicide, J,,. This is explained further in later sections.) The terms F, and Fik are the release rate of N atoms (atoms/cm’@) and the formation rate of the growing phase [formula units (f.u.)/cm’/s] for k phase of ith silicide, respectively. The terms A&, AC,: and AC,, are the driving forces for these processes to occur, and they are the difference in chemical potential of M atoms in the R region (J/atom), the free energy change for the release process per N atom and the free energy change for the formation process per formula unit of product. The first term on the right-hand side of equation (22) can be divided into two terms. i.e. AJA& and JuutA& :

I+- MS region --)1 If- St region ---+

ReactIon (R) region Fig. 3. Schematic of metal-Si diffusion couple which is the same as in Fig. 2, except the Si resion is divided into n slabs for the ournose of explaining I& concept if FEDR in a reactlon region. Each siabhas the

same thickness as the reaction region. (Not to scale.)

dsk - = AJA& + J<,,cAp; + r,AG,; + F,,AG,,, dt (23)

where AJ = JI,- J,,t is the flux of M consumed by the release and formation processes, while JO,, is the flux of M that flows through the R region but is not involved m reactions in the region. In the following discussion, competltion between reactions in terms of FEDR are to be examined. Since the competition is the main concern of this study and since the term J A&‘, does not contribute to the FEDR for release and out > formation processes, this term may be neglected for the purpose of this study. Therefore, equation (24) will be used to replace equation (22) in the following discussion :

dsk - = AJA&! + r,AG,,* + E;~AG,~. dt

Also, in cases where J,, B JO,,, J,, z AJ: and equation (24) will also be used.

In equation (24) the FEDR for the R region is expressed as a sum of three contributions with each part being a product of a flux and a force, which appears similar to the expression of local entropy production [equation (I 3)] in TIP theory. In the following section, these fluxes and forces will be discussed and the differences between equations (13) and (24) will be pointed out.

THERMODYNAMIC FLUXES AND DRIVING FORCES FOR SILICIDE REACTIONS

According to step 2 of our kinetic model, N atoms are released and form product “molecules” during the release process. The composition and chemical bonds of the “mol- ecuIes” are simiIar to those of the growing phase. The assumption implies that the “molecules” constitute the interface between the growing silicide and the non-moving reactant phase. It is also implied that steps 2 and 3 of the reaction take place at different locations of the R region, i.e. the release process occurs at the side adjacent to N, while step 3; the formation process, occurs at the other side of the region. For convenience, in the following the side where step 2 occurs is referred to as the N suvfizce, while that where step 3 occurs is referred to as the growing phase surfice.

Two characteristics of the R region are crucial to the derivation of thermodynamic fluxes and driving forces for the three- step process proposed by our model. First, the structure of the R region is poorly ordered or disordered and some “molecule” chemical bonds are not satisfied. This is mainly because the release of N atoms can be considered as an individual event, i.e. the release of one atom is independent of the release of others. Secondly, the composition is non-uniform throughout the region. It is expected that a relatively steep concentration gradient exists over the small width of the region (< 2 nm). The growing phase surface has a concentration of M equal to that in the growing silicide while the N surface is somewhat N rich, compared to the growing silicide. It will be seen that the energies arising from both the interfacial structure and the concentration gradient in the R region contribute to the fluxes and the driving forces.


D$fusion j&x and chemical potential d(fftirence in (I reaction where g,,(c) can be expressed as a sum of chemical potentials of region its constituents, i.e.

The thermodynamic flux and driving force for step 1 of sili- tide formation are the diffusion flux and the chemical potential difference of M between the two sides of the R region. Tu [14] has discussed the effect of concentration gradient on diffusion flux. The general expression for diffusion flux as a function of chemical potential gradient is [ 141:

go(c) = m,pLE: (c) +n,$(c). (30)

&(c) and pi(c) in this equation are the chemical potentials per M and N atoms (or per f.u.) in a solution of uniform concentration c. Combining equations (29) and (30) and applying partial derivatives to the resulting equation, according to the definition of chemical potential, one obtains

(25)

where D, c. i and ~F/.x are the diffusivity, concentration, average jump distance and chemical potential gradient of the moving species, respectively. For the condition where (-n(ap,lax)/k,T) << 1, J is a linear function of ap/Zx, i.e.

WJ)

I f p = ,~~+k,Tln c, then one obtains the usual form of Fick’s first law by inserting the expression for p into equation (26),

(27)

When a large concentration gradient. such as that at an interface region, exists, the condition (-i.(~~/dx),/k,T) cc 1 cannot be satisfied. In this case, the diffusion flux should be larger and expressed by equation (25). In practice, however, it is expected that the diffusion flux in the R region is limited by diffusion of the moving species in the growing silicide adjacent to the region and is equal to the diffusion flux from the silicide into the R region (i.e. J = J,,). Similarly, the flux out of the R region. J,,,, is limited by diffusion in the N lattice, so that the flux AJ can be determined by examining J,, and J,,,, in the two phases. respectively.

In order to derive an expression for Apz., the chemical potential difference in the R region, the expression for the free energy of an interface given by Cahn and Hilliard [134] can be used. According to the authors, the free energy, GK, can be expressed as

G, = AN,. [g,)(c) + E(dcid.u)‘] dx. (28)

The subscript R is added to indicate that this is the free energy of the R region in the following discussion. In the original Cahn and Hilliard equations, F and f are used instead of G and g, respectively. The first term in the integrand is the free energy per molecule of a solution of uniform concentration c. The second term in the integrand is the gradient energy per molecule, i.e. the free energy due to the presence of a large concentration gradient (dc/dx) in a small volume of non-uniform solution. The term & is the gradient energy coefficient, A and NV in the equation are the area of the interface region under consideration and the number density of “molecules” per unit volume of the region, respectively. From equation (28). the free energy per molecule is the sum of the two terms in the integrand.

g(c) = g,(c) +E(dc:‘dx)‘, (29)

T P n. = & [go(c) +~(dc/W’l,,,,

1 p:(c) +pE: (dcldx). (31)

This equation denotes that the chemical potential per M atom (,uF) in a small volume of non-uniform solution is a sum of two contributions [where the subscript R’ is used to differentiate ,L$ from $(c)]. One part is p!(c), the chemical potential in a uniform solution of concentration c. The second part is &‘(dc/dx), the chemical potential due to the contribution of “gradient energy”, which is an extra term and is usually negligible when the composition gradient is small. Similarly, the chemical potential per N atom, $ in the solution can be expressed as

pg. = p;(c) +p;: (dc/dx), (32)

where &(c) and pz(dc/dx) have the same meaning as ,&‘(c) and &‘(dc/dx), but are for N atoms. The chemical potential gradient across the R region, therefore, can be derived from equation (3 1) and is given by the following:

d/$ -zz dx

d&‘(c) + d&’ (dc/dx) dx dx (33)

Since the diffusion distance, i.e. the thickness of the interface .Y~, is so small the gradient can be approximately expressed as (A&/,+,), and

A& = ~~9 = A&‘(c)+A&‘(dcidx). (34) R

Equation (34) is the expression for the chemical potential difference in the R region, which indicates that the driving force consists of two parts, i.e. the chemical potential differences due to the composition difference and due to the “gradient energy” difference between the two sides of the region. It should be mentioned here that TIP theory has not given a general treat- ment for the cases where a large concentration gradient exists in the solution. The generalized expression of entropy production per unit time [equation (13)] is valid only when a large concentration gradient is absent. In other words, the contribution of the concentration gradient to the driving force of diffusion is neglected in that equation.

Maximunz releuse rute and ,fi-ee energy change for the relense process

Maximum releuse rate expression. The release rate expression as a function of diffusion flux, AJ, has been given in a previous paper [126] and is also shown in Fig. 4 :

L. ZHANG and D. G. IVEY: SILlClDE FORMATION IN METAL-9 DIFFUSION COUPLES 59

AJ -

Fig. 4. Schematic release rate vs diffusion flux plot

r, = rimax if AJ 3 Jiuc

r, = C,AJ if AJ < J,u,-, (35)

where ri is the release rate for the formation of ith silicide and rimax is the maximum release rate for ith silicide formation, corresponding to JiL.c, the upper critical flux for ith silicide formation. C, is the ratio of non-moving reactant to moving reactant atoms in ith silicide (the slope of the r vs J curve in Fig. 4). From equation (35). when AJ = JcUc, one has I’, = rtmdar and C, = rimax/J,UC, so that

r, = C,AJ = Y dJ i”ldX J (36) iuc

An expression for r,,,, will be derived in the following paragraphs from microscopic considerations. Consider a surface of an N (e.g. Si or a refractory metal) crystal lattice that is exposed to an R region in a metal-Si diffusion couple. The N atoms in the surface layer and the M atoms in the R region come into intimate contact. making chemical reaction between these atoms possible. If an N atom (or molecule) sitting on the surface has a vibration frequency Y, the frequency with which it can jump away from the surface into the R region is equal to v multiplied by a Boltzmann factor, exp { - E,,/k,T) Only those vibrations with energy higher than the activation energy, E,,, can free the atom from the bound state at the surface. Therefore, the maximum release rate of N for Ith silicide formation in this reaction region can be expressed as

(37)

where n* is the number density of N atoms per unit area of surface layer (atoms/cm’). The activation energy E,, for ith sihcide formation can be obtained by determining Y,,,,, at various temperatures experimentally and by drawing an Arrhenius plot. The activation energy E,; is smaller than the energy required to break the chemical bonds of N atoms, which is a very common phenomenon in chemical reactions. According to chemical kinetic theories of rate constants. this is because the simultaneous formation of a new chemical bond compensates for the breaking of the old bond [I%]. This point will be

considered in deriving an expression for E:,, (surface recon- struction is neglected for simplicity). When an M atom arrives at the N surface by diffusion, it will start to interact with N atoms surrounding it (Fig. 5). While the bonds between N atoms are being broken, new metal-Si bonds are forming. i.e. the two processes proceed simultaneously. The bond breaking process absorbs energy while the bond forming process releases energy. As soon as the chemical bonds between N atoms are completely broken, the metal-Si bonds are formed, although the bond structure is not exactly the same as that in the growing crystalline silicide lattice. The bond structure will be adjusted during the phase formation process following bond breaking. Therefore, the energy released due to metal-Si bond formation is only partially completed. Further energy release will continue as the bond structure adjusts. As a result, the energy required for breaking Si bonds is partially compensated by the energy released due to the formation of new bonds. This is schematically shown in Fig. 6. State 1, labelled in the figure, represents the state when M atoms arrive at the N surface and intermixing between M and N atoms occurs. State 2 indicates the energy state corresponding to the saddle point for the simultaneous bond breaking and forming process. State 3 in Fig. 6 represents the energy for the silicide molecule that forms during the process. State 4 at the peak of the solid curve represents the saddle point if no new bonds form during the bond breaking

(a)

(b)

Fig. 5. Schematic of chemical bond breaking and forming process in a reaction region. Closed circles and open circles represent Si and metal atoms, respectively. (a) Metal atom A just arrives at the Si surface and interacts with surrounding Si atoms; (6) Si-Si bond breaking and A-Si

bond formation occur at the same time.

L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL-Si DIFFUSION COUPLES

Reaction coordinate Fig. 6. Schematic energy vs reaction coordinate plot showing that the energy required for breaking Si bonds is partially compensated by the

energy released due to new bond formation.

process. The energy difference between states I and 2 is the activation energy E,,. The energy difference between states 1 and 3 is the thermodynamic driving force for the reaction. The energy difference between states 1 and 4, E, is the activation energy for breaking N bonds without new metal-Si bond formation, while the difference between states 4 and 2, E’, is the energy compensated by new bond formation during the N bond breaking process. Therefore, E,, is made up of two parts, i.e.

Eai = E-E’. (381

For a given reaction, E’ is proportional to the thermodynamic driving force for the release process, AC,, (see Ref. [I291 for detailed discussion about this relationship) :

-E’ = aAG,,. (39)

In the above equation, x is a proportionality constant. Inserting equations (38) and (39) into equation (37) yields

r ,“,ax = n*vexp [- yy].

The activation energy E, the vibration frequency \? and n* in equation (40) should be the same for a given reaction region, since the reactants are the same in all the reactions occuring in a given R region. In particular, the non-moving reactant is the same and the factors E, 1’ and n* are properties of the reactant. (However, E may change with crystalline orientation if N is a single crystal such as a Si wafer.) As a result, Y,,,, is exponentially dependent on the free energy change for ith silicide formation, AC,,, because according to equations (39) and (40), the energy compensated during the breaking of chemical bonds of one N atom is proportional to the thermodynamic driving force, AG,,, per N atom. In other words, a larger driving force results in a larger energy release for the bond breaking process and, hence, a smaller Arrhenius activation energy. This relationship reflects the unique character of silicide formation in metal- Si couples. It is this unique character that allows semiquantitative RPPs to be calculated using equation (40) and available thermodynamic data. From calculated RPPs, silicide formation sequences in 15 metal-Si systems have been successfully predicted [12X]. On the other hand, there are cases where the linear free energy relationship is not applicable. For example, when two phases compete with one another for nucleation. the phase with the larger driving force may have a larger activation energy than the other phase, if the former has a much larger surface

tension and/or a larger strain energy. This point will be discussed further in later sections.

Free energ?, change expression,for the release process. The driving force for the release process is the free energy difference between states 1 and 3 shown in Fig. 5. i.e. the state when M atoms arrive at the N surface and intermix with N atoms (state l), and the state when the N atoms are released into R region forming silicide “molecules” with the M atoms. Theoretically, the driving force per molecule can be expressed by the chemical potential of each component,

*.G,, = t& - w% - n,d&~ (42)

where & is the chemical potential of the product molecule in the R region, and #s and &!,s are the chemical potentials of the M atoms and the N atoms at the N surface (indicated by the subscript NS). The last two chemical potentials are not practically measurable, making equation (42) inappropriate for estimating the driving force in practice. An alternative approach is to express the driving force, AGi,, in terms of the standard free energy change (AC:) and the activity quotient Q,:

bTln Q, AG,, = AGP, + y. (43)

It can be shown, by means of a Gibbs free energy versus composition plot, that this is a reasonable approximation. In Fig. 7, the curve represents schematically the free energy of a silicide MSi and the straight line indicates the free energy of an ideal M-Si mixture at various compositions (where M represents the metal in the system). The free energy difference between points A and B is the standard free energy change AC:,,, = Gg- GA, where G, and GA are the free energy of pure MSi and the ideal M-Si mixture, respectively, at the same composition as that of MSi. I f the free energy change of mixing is considered, the free energy of the M-Si solution should be indicated, for example, by point C. Similarly, when the concentration of MSi at the N surface is taken into account (see discussion about Qt below), the free energy for MSi is lowered to, say, point D. As a result, the free energy change for transforming the M-Si solution into an MSi silicide (not pure MSi) is AGM,,,, = CD-Cc, which is equal to the AGMsl, expressed by equation (43). Two additional

G

GM

‘Si

1 I M MSi

Fig. 7. Schematic free energy vs composition plot. Points A-F represent free energy values G,. G,, G,, GD, G, and GF, respectively.


factors have to be considered. When metal atoms arrive at the Si surface, part of their chemical bonds is not satisfied while the chemical bonds of Si atoms are distorted by the intermixing. Furthermore, the concentration gradient between the surface Si layer and its neighboring layers is very large. These cause the free energy of this layer to be higher than that for the uniform bulk M-Si solution. The difference in free energies of the Si surface layer and the M-Si solution of the same composition is ACM,,,, = GE- Cc, which is equivalent to an interfacial energy in terms of dangling or distorted bonds and a concentration gradient. When AG,s,s is added. the free energy of the Si surface layer is indicated by point E. Similarly, the newly formed “molecules” in the R region are not in a uniform solution either. Therefore, the free energy of the “molecules” in the R region should be higher than that at point D and are, for example, represented by point F, i.e. the energy difference G,- Go is the interfacial energy of the R region. Consequently, the actual free energy change for the release process should be GF - GE, which can also be expressed by equation (42). However, since GE is unknown, GF- GE can be estimated as AG,s,, = G,-G, [i.e. using equation (43)], although some error will be introduced. This is a reasonable approximation, since it may be expected that in most cases the error would be small when compared w-ith the driving force for the reaction.

An expression for the activity quotient [Q, in equation (43)], which takes into account concentration and defect effects, has been derived in previous papers [127-1291 and therefore only the equation is given here.

Xy, xy” and X, are the mole fractions for the reactants (N and M) and the product, respectively. Insertion of equation (44) into equation (43) yields the following expression for AG,,:

AC, = AC: + y ln { (x,):[(x~)n,(x~)~z,]~. (45)

This equation is useful when a comparison of driving forces causing different silicides to form is required. From this equation, the comparison is based on the free energy change per non-moving reactant atom (or per mole) which is released. Therefore, a silicide with more than one N atom in its molecule (or f.u.) may have a smaller AC,, because it, > 1 [note that AC: in equations (43) and (45) is already defined as the standard free energy change per N atoms and is equal to the standard free energy change per f.u. divided by n,]. A rilicide molecule with more than one non-moving reactant atom has a lower likelihood of forming because its formation requires more non- moving reactant atoms to he released at the same time. This phenomenon has been considered in equations (40), (43) and (45) by expressing E and AC,, as an activation energy and a free energy change for releasing one N atom.

Maximum formation rate und free energy change ,for the ,formation process

Free energy change qfformation. The driving force for formation, i.e. the process of rearranging product “molecules” on the surface of a growing phase k, is the difference in free energies between the product “molecules” in the reaction region and

those on the surface of the k phase. The driving force can he expressed by an equation similar to that used by Cahn and Hilliard [ 1341:

AC,, = m,&’ +ni$ -g,(c) -E(dc/dx)‘, (46)

where the last two terms have the same meaning as those in equations (28-30). The terms py and ~2 are the chemical potentials per M and N atoms in the k phase, respectively. Inserting equation (30) into equation (46) yields

AG,k = m,&‘+n,$ -m,&‘(c) -n+;(c) -F(dc!dx)’ or

AG,k = Ag--(dc/dx)‘. (47)

m,&(c) and n,&(c), as defined in equation (30), are the chemical potentials per M and N atoms in a uniform solution of composition c in the R region. As a result, Ag in equation (47) is the free energy difference per molecule due to the differences in structures and compositions between the “molecules” in the growing phase and those in the R region. The value of AG,n can be estimated from the surface tension, ckkN, of the interface between the k phase and the non-moving reactant phase (which is the R region during k phase growth). According to Cahn and Hilliard,

-AC,, tot OkN = ___ = A

-NV [Ag--(dc/dx)‘]dx> I

(48)

where - AGzk ,“, is the interface energy of area A. The minus sign in this equation is used because the terms in the integrdnd are arranged in such a way that AGtk is negative when deriving equations (46) and (47), but the surface tension crkN should have a positive value. From this equation, the average driving force per molecule, AG,k, can be obtained from

~,z%=L x,N,, xR

[Ag-E(dc/dx)2] dx, (49)

where xR is the thickness of the interface. Cahn and Hilliard [134] have proposed an optical method to measure xR for metal samples. By using HRTEM. xR is also measurable for other crystalline samples. Therefore, the average driving force per molecule for formation can be determined if okN is available. At this point. it should be mentioned that equations (47) and (49) cannot be obtained from TIP theory since the driving force involves an interfacial process and contains a “gradient energy” term due to the existence of a large concentration gradient.

Figure 8 shows schematically how the free energy is dis- tributed over the R region. In the figure, G, represents the free energy of the N phase. GE, GF and G,, are the free energies of the N surface, R region and the growing phase (k): respectively. The NS region is the N surface layer. When M atoms diffuse to the N surface, some intermixing occurs and the free energy is lowered from G, to G,. As soon as N atoms are released into the R region and form new “molecules”, free energy, GF - GE, is released. The GS region is the surface of growing phase k. The ‘*molecules” are driven to the region to crystallize by the energy difference, G, - G,, which is equal to the interface energy per “molecule”. As the reactions proceed. the R region and, hence, the free energy distributions over the region will move

62 L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL--% DIFFUSION COUPLES

GS r&w” x---w

Fig. 8. A schematic showing the distribution of free energy within the reaction region. G,, G,, G, and G, are the free energies of the non- moving reactant phase, N surface, reaction region and growing silicide. respectively. As the reactions proceed, the free energy distribution curve

moves to the right of the figure together with the reaction region.

together to the right of the figure. leaving crystalline silicide, i.e. the k phase, behind.

It is possible to have more than one form or phase for a given silicide (e.g. an amorphous phase and a crystalline phase of the same composition). In this case, the driving forces for different phases of the silicide are not the same. This is schematically shown in a Gibbs free energy vs composition plot in Fig. 9. There are, for example. two possible phases for the ith silicide. the stable phase (il) and the metastable phase (i2). In the figure, G, is the free energy of “molecules” in the R region, G,, and G,* (i.e. Glk = m,&’ +n,&) are the energies of the “molecules” in stable and metastable configurations, respectively. Therefore. the driving forces for the two phases are AGi, = G,, -GF and AC,? -G,. The stable phase has a larger driving force.

An interesting point that may be drawn from the above result is that if the free energy GI1 for a metastable phase is equal to or larger than GI;, there is no driving force for this phase formation. This can help explain why a large heat of mixing for a metal-% system does not guarantee amorphous phase formation (see Ref. [9] and the Introduction in this paper for comments on solid state amorphization). In this case. the heat of mixing is considered as the thermodynamic driving force for solid-state amorphization (SSA) [9. 136, 1371, i.e. the enthalpy

G

Composition Fig. 9. Schematic free energy vs composition plot. G,, and G,, indicate the free energies of the stable phase and metastable phase of the ith silicide. respectively. G, represents the free energy of the Ith silicide molecule at the reaction region (reactive interface). AC,, and AC,* are the driving forces for phases I and 2 to form, respectively. The stable

phase has a larger driving force.

of the amorphous phase. H,,, can be approximately taken as G,? in Fig. 9. As a result, no matter how large the heat of mixing is, there is no driving force for an amorphous phase to form if H am x G,2 > G,. As a result, a crystalline phase, rather than an amorphous phase, will grow.

Maximumformution rate. Expressions of formation rate as a function of release rate have been given in our previous papers [126, 1291 (also demonstrated in Fig. 10):

Fik = z if r, < Y,~

Fgk = F,: n,ax if ri > rik, (50)

where nj is the number of N atoms per formula unit of ith silicide and rik is a critical release rate related to the maximum formation rate. F,kmdrr for k phase formation (if ith silicide has more than one phase or form. k = 1, 2 . is used to indicate different phases). When the release rate is low? F,, is a linear function of r,. I f Y, is equal to or higher than the critical value rikZ F,k reaches a maximum value and becomes constant. The maximum formation rate, Fjkmax, is determined by the energy barrier for the formation process of the growing phase k. The energy barrier is the barrier which must be overcome when the chemical bonds and the molecule coordination and composition in the R region are adjusted to those in the k phase. Therefore, F,, rnari is independent of release rate r,. At the initial stage of k phase formation. however, the nucleation of the phase has not been accounted for. In a formation rate versus release rate plot (i.e. F vs Y plot, such as in Fig. lo), the horizontal line labeled as F,, indicates a formation rate that is equal to the nucleation rate of k phase. Since nucleation is a necessary condition for a phase to grow in most cases, Flk. represents an upper limit for Fik. The term Fik. can be referred to as the conditional maximum formation rate for k phase. Correspondingly, the release rate ril. is a conditional critical release rate:

F,kEF,a.=x if r,>r,, 12,

F =; ii if r, < Y,~,.

n.

x-l

I

I Fi2'

(51)

F

r Fig. IO. Schematic formation rate (F) vs release rate (r) plot. Fzk. represents the conditional maximum formation rate for phase k of the ith silicide; T,~. indicates the critical release rate corresponding to the F,,

L. ZHANG and D. C;. IVEY: SILICIDE FORMATION IN METAL -Si DIFFUSION COUPLES 63

If, at the very beginning of the reaction, the release rate I’, is smaller than rikr the k phase will nucleate. Otherwise, another phase of ith silicide may nucleate first.

It has been reported in the literature that in some metal- Si systems, metastable phases, particularly amorphous phases, grow before the stable phase can form. This means that these metastable phases have larger maximum formation rates, Fzkman. According to the discussion in the last section. the driving forces for metastable phases are definitely smaller than those for stable phases in each respective system. It is also noted that in many metal-Si systems, no metastable phase formation is observed. These phenomena indicate that a larger driving force is not necessary to correspond to a larger Flhmal. Therefore, an expression similar to equation (40) for rrman cannot be expected for Fikman. This makes the expression for F,,, mill quite simple, i.e.

(52)

where v, and nk are the vibration frequency and number density (f.u./cm’) of ith silicide “molecules” per unit area of reaction region, and E,: is the activation energy of rearranging a molecule from the reaction region to the growing phase. Elk is the energy required to adjust the bond structure coordination of the “molecules” in the R region to that of the growing phase. It is conceivable that as less adjustment is required, & becomes smaller. For example, the amorphous phase has the most open and least ordered structure compared with crystalline phases of a given silicide. In addition, its structure may also be closest to that of the R region. Therefore, an amorphous phase may have the smallest E,, and therefore the largest FIL,,,,,. Also, if a crystalline phase has a complex structure which is significantly different from that of the R region, E,A for this phase will be large, and, consequently, Fs:lkm.,x will be small.

In practice, a conditional maximum formation rate. F,,, may be more significant. since nucleation is the first step for a phase to grow. From equation (52), F,, can be derived easily as follows:

Flk. = F,, max exp (53)

The term exp [-AG,i/k,q in equation (53) is the Boltzmann factor for k phase nucleation; AC,; is the energy barrier for heterogeneous nucleation. According to classical nucleation theory, if a nucleus in the shape of spherical cap is assumed,

AG:, = AC:,,,, x 2-3cosB+cos’O

4

In this equation. 0 is the contact angle and AC,,,,, is the energy barrier for homogeneous nucleation :

AG’ -!!?! (An)’ lk hom - 3 (N,, AC,, + AHJ2 ’

where Ao is the difference in interface energies due to k phase nucleation, Njk is the number density of “molecules” per unit volume of k phase, and AH, is the elastic strain energy per unit volume, induced by nucleation. Equations (54) and (55) are the same as those given by classical nucleation theory, except that

the driving force, AGik, the free energy difference between the k phase and the R region per unit volume, is different from that used in classical nucleation theory, i.e. AC, the chemical free energy change per unit volume. AG, includes all the free energy changes for the entire process. since an interfacial reaction in a diffusion couple is conventionally considered as a one-step process. This concept is applicable to the case where the interface between a growing phase and a contracting phase (made up of the nonmoving reactant atoms or “molecules”) is coherent or semicoherent, which will be discussed later. In other cases where the interface has a thickness of a few atomic or molecule layers, it is unlikely that the chemical reaction between reactant atoms and the transformation of the same atoms from the interfacial region into the growing phase can occur at the same time. Therefore. using AC,, to describe the driving force for the formation process is more reasonable. It should be emphasized that, from equations (53))(55), the activation energy for the conditional maximum formation rate, F,,., is a function of four variables, i.e. AG,k, AQ, AHd and 0. Since the latter three variables can vary from phase to phase, the linear free energy relation (LFER) for rrmaX is not applicable to Fik.. A large driving force does not guarantee a small activation energy and a large Fik This feature of FIk. makes metastable phase formation possible.

By combining equations (51) and (53), the formation rate can be expressed as a function of release rate and activation energies for formation and nucleation, i.e.

F,, = Fsk,,,exp - s [ ‘I

if r 3 rri.. (561 B

CRITERIA FOR SILICIDE FORMATIOlV

Driving force as a function &%LX

Expressions for the driving forces, i.e. A$., AG,,. and AC,, for moving reactant diffusion, non-moving reactant release and k phase growth. have been given in equations (36), (43) and (47), respectively. It is conceivable that the driving forces will change as the reactions proceed. Since the reactions are induced by diffusion and since a smaller diffusion flux means less matter is involved in the reactions, the driving forces may decrease as a result of a change in diffusion flux. In this section, the changes in driving forces will be examined. Equations (36) and (40) can be combined to give:

= n*vexp E+ aAG,,

-___ k,T

From equation (57), a new free energy term, AC,,*, is defined such that the term in (AJ/J,,J is combined with AC, by means of equation (45).


ctAG,,? = ctAG,, - k,Tln (AJ/J,,c) AG,k = Ag-E(dc/dx)’ < 0 if ri > 0,

= xAG; + Fin j(Xy)i[(X,“)“,(AJiJ,~~)n,~~(,~~)n*,]~. and

so that

AC,,* = AC: + Fin {(X:~[(X:)~,(AJIJ,,,)‘,~~(~~)~‘,]~. (58)

When AJ is equal to or exceeds Jiuc [equations (35) and (36)], then AJ/J,,, is taken to be equal to 1, so that AC,,* = AGi,. When AJ < J,m AG,* is greater than AG,, (i.e. -AC,,* < -AC,,), and as AJ decreases, AG,: increases. Equation (57) can be expressed in terms of AG.* i.e. it- 1

r, = n*vexp E+ xAG,,*‘(

- k,T j

According to the derivation of equations (40) and (58), and previous discussion, AC,,* is the driving force for the release process and is a function of diffusion flux.

Unlike AG,,*. the driving force for the formation process, AG,k, does not decrease continuously. because of its unique character. According to Cahn and Hilliard [I 341, the thickness of an interface is not an independent variable once the temperature and pressure of the system are specified. Thus, under the conditions used in this analysis, i.e. the system is at constant temperature and pressure, the interfacial layer (reaction region) thickness xR is a constant, x,., corresponding to a minimum interfacial free energy. As a result, any change in xR would increase the interfacial free energy and the interface would act in such a way that the minimum energy state would be recovered. During a release process, r, > 0. so that xR will increase. The interfacial free energy as a function of thickness xR is. according to equation (48),

-AG,/,,,, = -AN, i

” [Ag - E(dc/dx)‘] dx, (601 ”

where the 0 indicates the origin at one side of reaction region and xR is the thickness of the region. The rate of interfacial free energy change with respect to xK at constant temperature and pressure is

(61)

where [Ag-/?(dc/dx)2] is an implicit function of xi+, Since the interfacial free energy (- AGrk ,,,,) is a minimum at xR = x, , the derivative on the left hand side of equation (61) should be equal to 0 at a thickness of I, and be larger than 0 when xR > x,. Therefore:

Ag-&(dcjdx)’ = 0 at xR = x,

Ag-K(dc/dx)’ < 0 if xR > x,. (64

During silicide reactions, when r, > 0, more and more product “molecules” are formed in the R region, which tends to increase the thickness of the region, i.e. xR > x,. When ri = 0, xR = x,. From equations (47) and (49), one then obtains:

AC,, = Ag-L(dc,/dx)’ = 0 if r, = 0. (63)

From equation (63), it can be seen that during a silicide reaction, as long as newly formed silicide “molecules” are released into the R region, they will be driven out of the region by the thermodynamic force AG, and rearranged on the growing k phase. Once diffusion of M atoms stops, i.e. AJ = 0, so that r, = C,AJ = 0, the driving force will also vanish. For cases where AG,k < 0 and r, < 0, since the interface thickness is kept at x, to comply with the minimum interfacial free energy condition, the driving force, AG,k, can be considered to be a constant and approximately equal to the average value of the interface energy per molecule. This provides a measurable physical quantity as long as the surface tension for this interface is known [see equations (47)-(49)].

An analytical expression for the chemical potential difference, A&, i.e. the driving force for diffusion, as a function of diffusion flux has not been developed in this paper. However, some general tendencies regarding changes in Apg corresponding to flux J can be qualitatively discussed. The magnitude of A& is mainly determined by the chemical potential difference between a growing silicide phase and a non-moving reactant phase and by the M concentration at the N surface. Therefore. A&? does not change (decrease) significantly during a reaction for a given silicide phase; however, A,L$ usually does decrease with the formation of each new silicide. For example, in a near-noble metal-Si diffusion couple, the most frequently observed first silicide to form is M,Si (where M represents Ni, Pt, Pd, Co, etc.), which is followed by MSi and then by MSi2 (for NiiSi and Co-% systems only). Obviously, the chemical potentials of metal atoms are smaller in the silicide with a lower concentration of metal. Therefore, each time a more Si-rich silicide forms, A$ at the growing phase/Si interface becomes smaller than it was prior to the new silicide growth. When the system reaches thermodynamic equilibrium, both AJ and A& are equal to zero, because the chemical potentials of M are equal to the same value throughout the system. It is also note- worthy that if the system is not in an equilibrium state while AJ z 0, such as a system held at room temperature, A$ still has a non-zero value. Therefore, this driving force is different from AG,,* and AG,k for release and formation processes, These two forces vanish whenever diffusion stops.

From equation (24), the FEDR per unit area of reaction region, dga/dt, is

&A - = AJAt$ + r,AG,,* f F,,AG,k. dt (24)

The expressions for A&, Fik, AG,: and AG,k are given by equations (34). (56). (58). (59) and (63), respectively. There are two essential conclusions that can be drawn from these equations and the preceding discussion in this paper. First, with the advance of a reaction in a given R region, a decrease in diffusion flux AJ with time will result in a corresponding decrease in the release rate and formation rate, while the driving force for the release process will also decrease (AC,,* increases). The driving forces for diffusion and formation processes, i.e. A& and AGikr


are almost constant for a given phase formation pi-ocess. There- fore, the total FEDR in the R region will continuously decrease with time. Secondly, when thermodynamic equilibrium is estab- lished in a given system (e.g. a Si-richest phase for a thin metal film on a Si substrate), all the fluxes AJ, Y. and F and all driving forces go to zero so that dgkidt = 0. These two conclusions are in agreement with the general results from TIP theory, i.e.

(64)

where P = d,S/dt is the entropy production per unit time. Equa- tion (64) indicates that if a system is in a stationary state (according to TIP, a thermodynamic equilibrium state is considered as a special case of stationary state), the time variation of the entropy production (dP/dt) is equal to zero. In this state, entropy production per unit time is a minimum. In particular, when a stationary state is the thermodynamic equilibrium state, d,Sjdt = 0, it certainly satisfies equation (64). When the system is away from stationary state, the time variation in entropy production, dP/dt, is less than zero. which means the system is irreversible and entropy production decreases with time. The two conclusions mentioned above can be expressed as

The sign 3 is used because d,Gk/dt has a negative value for irreversible processes. The terms d,Gk/dt and A in this equation represent the FEDR in the entire reaction region and the cross- sectional area of the region, respectively. This equation is significant because it indicates that when the solid state reactions in a diffusion couple advance toward thermodynamic equilibrium. the FEDR for each R region increases (i.e. - d,Gk/dr decreases) with time. However, neither equation (64) nor equation (65) specifies how the entropy production per unit time decreases and the free energy degradation rate increases with time. In the following section, it will be shown that in an R region and at a given diffusion flux (AJ), there are always some reactions (release and formation processes) that may result in the largest FEDR in the region. From these results, the criteria for silicide formation are proposed.

Criteria.for silicide.formation

From equation (24). the FEDR for the release process. dg8,/dt, can be expressed as follows:

(66)

Inserting equation (59) into equation (66) yields

If in an R region there are three possible release rates and Y> > r, > r3 at AJ,,, as shown in Fig. 4, one obtains (AG:I > IAG:j > lAGPI by applying equation (59) (n*, r, and E are the same for all reactions in a reaction region). Inserting these results into equation (67) gives

(68)

which indicates that the largest release rate (rL) will produce the largest FEDR.

From equation (24). the FEDR for the formation process, dgtk/dt, can be expressed as

Figure 10 shows a schematic formation rate versus release rate plot for three phases of the ith sihcide. Two possible cases can be considered. In the first case: phase 1 in the figure is stable while phases 2 and 3 are metastable. The relationship among the driving forces for formation is: AG,, < AC,? < AGi3. At a given release rate, r,,. it can be seen from Fig. 10 that F,, >

F,,, > F;,., i.e. the formation rate of phase I is larger than the conditional maximum formation (nucleation rates) of the other two phases. Applying equation (69) to each process results in the relationship

or

(70)

which means the growth of phase 1 will produce the largest FEDR in all possible processes. When the release rate decreases and passes a critical release rate rrZ.. then F,, = F,2. > F,?,. Even in this case, equation (70) is still valid because phase 1 has the largest driving force. Therefore, the formation of the stable phase will result in the largest FEDR throughout ith silicide formation.

In case 2, phase 2 in Fig. 10 is stable (phases 1 and 3 are metastable) and AG,? < AG,, < AC,;. According to Fig. IO. at rOr F,, > F,,, > F,,. I f F;, is significantly larger than F,, , the relationship between the FEDRs for these processes is

Fi,AG,, < F,:AG,z -=c F&3,,. (71)

This is because the driving force for the stable phase (phase 2) is usually only a few times larger than that for the metastable phase (see previous discussion), while F,, is much larger than E;,,. When the release rate decreases as the reaction proceeds, at some point F,, AG,, will be equal to F,* AG,L. After this point, the relationship expressed by equation (7 I) will change to

F,AG,z < F,,AG,, < F;,AG,,. (72)

In case 2, therefore, the metastable phase (k = 1) will induce the largest FEDR when its formation rate is much larger than that for the stable phase. As soon as equation (72) is satisfied due to a decrease in the release rate, stable phase formation will result in the largest FEDR. Similarly, the case when phase 3 in Fig. 10 is stable. while the other two phases are metastable, can be explained as above.


If the ith silicide has the largest release rate in an R region and k phase growth of this silicide induces the largest FEDR in all possible formation processes. the total FEDR due to the formation of k phase of ith silicide in the R region is also the largest compared to all other reactions, i.e.

,j = 1,2.3 . and j # i. (73)

So far the largest FEDR for the diffusion process has not been examined. Although competitive diffusion processes do exist in bulk and thin film diffusion, the competition in an R region is negligible for three reasons. Firstly, the concentration gradient in the reaction region is much larger than that in other diffusion regions. Secondly. the interface structure is more open than that of other solid phases. Thirdly, the diffusion distance is rather short, only a few atom layers thick. As a result, diffusion in this region is very fast. It is expected that the diffusion flux in the R region (J) is determined by the diffusion flux (J,,) from the growing phase into the R region. Therefore, J,,A&, represents the largest FEDR for diffusion in the R region. For the purpose of comparing the FEDRs of all possible reactions, the FEDR due to diffusion in the R region can be expressed by AJA&, which is actually the same for all possible reactions. (Also, see equations (22)-(24) and discussion thereof.)

What, then, are the implications of the largest FEDR in terms of the silicide formation sequence? The significance of the largest FEDR can be shown in Fig. 11, a free energy versus time plot. In this figure, the curves represent the possible paths of free energy change for a given system. Curve 1 indicates the path of the largest FEDR. It is obvious that at the time instants just beyond the tangent points P3 and P2, respectively, the slope of the tangent for curve I has a larger negative value than those for the other curves. Since the slope of each curve corresponds to the FEDR at a given time, the largest FEDR means that no curve deviating from curve 1 can have a free energy value lower than those of curve 1. Therefore, curve 1

Fig. II. Schematic free energy vs time plot. Curve 1 represents the reaction path with the largest FEDR. Points PL and P, are the respective

tangent points where curves ? an d 3 start to deviate from curve 1.

represents the lowest free energy states of the system at any given time, which can be referred to as relative minimum free energy states. When a system advances along a path of the largest FEDR, it can decrease its free energy in the fastest way and hence keep itself at the relative minimum free energy state until it reaches the “absolute” minimum free energy state, i.e. the equilibrium state. Since at any time along the path the relative minimum free energy state is the most stable compared with all other possible states, the largest FEDR path is com- patible with the system’s final state, i.e. the equilibrium state. Also, it is not contradictory to the general results of TIP theory expressed in equations (64) and (65). as long as the slope of the curve, or the FEDR. coniinuously increases and becomes zero when the system reaches the equilibrium state. Therefore, it is reasonable to consider the largest FEDR path to be most favorable for irreversible processes.

From the results, a criterion for solid-state reactions in metal- diffusion couples is proposed: during silicide reaction in a reaction region of a metal-Si diffusion couple, a number of possible reactions always compete with one another. The reactions which result in the largest FEDR will actually occur. The criterion is referred to as the largest FEDR criterion.

When the criterion is applied to the release process, it can be translated to a simple kinetic criterion: among all possible release processes at any diffusion flux, the one with the largest release rate will actually take place. For formation processes at a given release rate of ith silicide, the phase with the largest (- F,,AG,,) will form.

DISCUSSION

In the preceding sections, the kinetic and thermodynamic factors which control solid-state reactions in transition metal- Si diffusion couples have been closely examined by means of the TIP theory, Cahn and Hilliard’s approach to interfacial free energy, and our kinetic model. The main results are summarized below.

From basic principles of TIP theory [equations (2)-(9)]. it has been shown that the free energy change due to silicide formation in a diffusion couple can be determined by examining the free energy change of the R region located between the growing silicide and the non-moving reactant phase, The free energy degradation rate per unit area of a given R region can be expressed [equations (22)-(24)] as a sum of three contributions, each corresponding to one of three steps, i.e. moving reactant diffusion, non-moving reactant release and product molecule crystallization. Each term is a product of a thermodynamic flux and a driving force. These fluxes and driving forces have been examined individually. A microscopic mechanism is assumed for the release process, i.e. one in which the N atom bond breaking process and the new product molecule bond formation process occur simultaneously. It is shown through the use of a schematic plot (Fig. 6) that the driving force for the release process can be approximately expressed by equation (43), so that the value of the driving force can be estimated if the defect density of non-moving reactant surface [equation (45)J is available (those interested in this topic are referred to Ref. [128]). For a formation process, the conditional maximum formation rate, F,L,, is probably of more practical importance. The term

L. ZHANG and D. G. IVEY: SILICIDE FORMATlON IN METAL-3 DIFFUSION COUPLES 67

FjLj is exponentially dependent on two contributions to the activation energy [equations (52) and (53)], one for nucleation and the other for rearranging product “molecules” from the R region on to the crystalline lattice of the growing phase. The expression for AG,L, [eqwdtiOnS (54) and (55)], the activation energy for ik phase nucleation, is basically the same as that given by classical heterogeneous nucleation theory. except that the definitions of the driving force are different. The driving force, AGik, for the formation process [equation (47)], is derived based on step 2 of the new kinetic model and Cahn and Hil- hard’s expression for interfacial free energy. Its value can be approximated, using equation (49). from the surface tension crkh of the same interface. In particular. it has been shown that the driving force AC, is related to the release rate [equation (63)]. When r, > 0, AGik is almost a constant, while for r, = 0: AG,k = 0. This is because the interface tries to maintain a specific thickness which corresponds to a minimum interfacial free energy at constant temperature and pressure. An expression for AG,*, the driving force for the release process, is also given by equation (58). From this equation, AC,* = AC,,, when the flux is equal to or larger than J,,c, and AC,* increases if the diffusion flux decreases. By qualitatively analyzing how these fluxes (J, r and F) and forces (A&, AC,: and AC,,) change with time, it is apparent that the FEDR of the R region will decrease with time [see equation (65)]. This result shows that the expressions for FEDR in the R region are in agreement with the general results from TIP theory [equation (64)]. From this point of view, it is further shown that in an R region, there is always a reaction that will result in the largest FEDR in this region. Moreover, it is shown through a schematic free energy versus time plot (strict mathematical derivation would be ideal, but is not available at this time and is beyond the scope of this paper) at any instant in time, that the largest FEDR leads the system to a relative minimum free energy state that is most stable compared with any other energy state at that instant. Therefore, a criterion for silicide reaction is proposed.

It should be emphasized at this point that, although the expression of the largest FEDR criterion proposed by the authors looks similar to that given by Bent: [9]. there are significant differences between the two criteria. The differences are addressed in the following.

(1) The concept of FEDR was first proposed by Ben&; however, it was not well defined in Ref. [9]. It is not clear whether FEDR referred to the free energy degradation of reactive interface only or to that of the whole diffusion couple under consideration. In BenC’s expression for FEDR, - DAG. D is the diffusion coefficient of the growing phase, such as initial amorphous phase, while the AC is the free energy change due to the growth of this phase [9]. In the present paper, the concept of FEDR has been redefined as the Gibbs free energy decrease per unit time due to changes within the reaction region (or reactive interface), d,G,/dt. A mathematical expression for the term d,G,/dt has also been developed. Moreover, it is shown for the first time that a knowledge of the FEDR due to changes in the reaction region is sufficient for examining silicide formation which takes place at the reactive interface in a diffusion couple.

(2) The approaches for the two criteria are completely

different. Ben& formulated his criterion mainly by the analogy of “metastable chemical structure“ that forms in metal-Si couples with the metastable shapes that form during dendrite growth and snowflake formation [9]. The approach in the present paper is a combination of TIP theory, Cahn and Hilliard’s approach to interfacial free energy [ 1341, and our recent kinetic model. A number of new results, such as definitions and expressions for the thermodynamic fluxes and driving forces for each reaction step. cannot be derived without using the new approach. Furthermore, this approach enables us to show, for the first time, that when a number of possible reactions compete with one another in a reaction region, there is always a reaction that will result in the largest FEDR in this region.

(3) The criterion from the present paper has a much wider scope than that of Bent-. In Ref. [9], Bene’s criterion was used only to explain silicide nucleation. more specifically, the initiation of amorphous phase and the first crystalline phase. The expression for the criterion could only be applied to qualitatively explain diffusion controlled processes. The newly proposed criterion combined with the recent kinetic model is able to explain the whole silicide formation process in a given metal-Si couple, i.e. from the initial reactions at the metal-Si interface to the final equilibrium state of the couple. If the necessary thermodynamic and kinetic data are available, this criterion should be able to predict the reactions quantitatively. It does not matter whether the reactions are diffusion controlled. nucleation controlled or interfacial reaction controlled. as long as the assumptions for the kinetic model and those for the derivation of the FEDR expressions are satisfied. Moreover, it is expected that the criterion, or at least the principles for deriving FEDR expressions, can be applied to bulk metal-Si diffusion couples and other binary systems (such as metal-metal couples) as long as a suitable kinetic model for the given systems is available.

In the following paragraphs, the above model coupled with our previous work on reaction process plots [ 126-1291 will be applied to discuss some practical problems.

According to our recent model, the stepped curve in an I’ vs J plot (e.g. Fig. 4) indicates the path of release processes in a reaction region. Since it represents the largest release rate at any given diffusion flux, the curve can be used to predict silicide formation sequence in this region. It can be predicted, theoretically, that any silicide in the equilibrium phase diagram of a metal-Si diffusion couple can form first if the release rate for this reaction is on the stepped curve of the r vs J plot, and if the initial diffusion flux is between J,, and J,,, the lower and upper critical fluxes, respectively, for this silicide. On the other hand. if none of the release rates for a silicide to form is on the stepped curve, the silicide is unlikely to form through direct metal-Si reaction. See, for example, Fig. 12, where an r vs J (or AJ) plot is drawn for a hypothetical reaction region. There are three possible silicides in this system, but only M$i and MSi, lie on the stepped curve; MSi does not. According to our model, MSi cannot form initially, only M&Si or MS& depending on the diffusion flux. Furthermore, it is also possible to predict which

68 L. ZHANG and D. G. IVEY: SILICIDE FORMATION IN METAL-Si DIFFUSION COCPLES

*Jo AJ

Fig. 12. Schematic r vs A/ plot showing release path for a three-silicide (M$i, MSi and MSi?) diffusion couple.

new silicide will form in a diffusion couple and when, providing Y vs J plots for this couple are available. Using the plot shown in Fig. 12, if the initial flux is AJ0 then MzSi will form first. The diffusion flux continuously decreases as the first silicide grows. Whenever the diffusion flux reaches a critical value, Jut (i.e. a plateau in the curve), a new reaction with its own release rate related to this Jut is initiated and a new silicide starts to grow (MSi, in this case). This approach has been successfully applied to initial silicide formation and multiple phase growth sequences in fifteen metal-Si systems [128], which in turn has provided strong support for the proposed criterion.

At present, quantitative calculations of r vs J plots cannot be achieved because of the unknown factors, E and c(, in equation (40). It is possible to determine these values from careful diffusion couple experiments; however. the discussion of this is beyond the scope of this paper and better dealt with separately. Semiquantitative Y vs J plots can be calculated from heat of formation data. as discussed in Ref. [128], and predictions can be made from these.

According to the criterion for formation processes in the last section. the formation of a phase can be predicted using an F vs Y plot (Fig. 10) and the data for AGik. At the present time, however, the capability of the prediction is limited because very little is known about actual F vs Y plots and the driving force for the formation process. AG,,. Therefore, the criterion and other results for formation processes from the preceding sections can only be used to explain qualitatively the experimental phenomena reported in the literature at this time.

From step 2 of the kinetic model and the expressions for Y,,,, and F,k, the release process occurs at the N surface while the formation process takes place at the growing silicide surface. The energy change as a function of reaction coordinate can be schematically shown by the solid curve in Fig. 13. The numbers 1,2, and 3 indicate the energy states before and after the release process, and after the formation process, The energy differences between states 4 and 1 and states 5 and 2 are the activation

E

h

Reaction coordinate Fig. 13. Schematic plot showing activation energies for interfacial reactions taking place at coherent and incoherent interfaces, respectively. Numbers 1, 2 and 3 indicate the energies corresponding to the states before and after N release, and after formation, respectively, for a three- step process at an incoherent interface. The free energy differences between states 4 and 1 and states 5 and 2 are the activation energies for release and formation processes, respectively. The energy difference between states 6 and 1 is the activation energy for the reaction occurring

at a coherent interface.

energies for the release and formation processes, respectively. This curve is not valid for the silicide reactions occurring at coherent or semicoherent interfaces. where the release and formation processes actually take place at once. Therefore, the reaction rate (in order to differentiate it from the terms “release rate” and “formation rate”, the term “reaction rate” is used to discuss the rate of this reaction and Y is used to denote it) is expressed as follows:

Y = n*vgexp aAG, + E, + AC:,

- I k,T . (74)

IUC

The terms AJiJruc, n*, v and a have the same meanings as defined previously; AG, = AG,+AGil is the total free energy change of the reaction; and AG1i is the energy barrier for nucleation but is small in this case because of the low interface energy at the coherent interface. The term Fi is a sum of two contributions, i.e. the energy barrier for breaking non-moving reactant bonds and the energy barrier for adjusting the bond structure of the product “molecules” onto that of the growing phase lattice. The latter may be even higher than that for the three step process, because at a coherent interface the adjustment and hence the movement of atoms are restricted due to the coherency requirement. In addition, it is likely that the strain energy is higher at a coherent interface than an incoherent interface and it will increase as the reaction proceeds, which will also contribute to activation energy, E,. As a result, E, will be significantly larger than either E for the release process or Eik + AG,; for the formation process in a three-step reaction. This is shown by the dashed curve in Fig. 13. Numbers 1 and 3 indicate the energy states before and after the reaction; the difference between 6 and 1 represents E,. Consequently. Y will be much smaller than the release rate and the formation rate in a three-step reaction for the same silicide forming at an incoherent interface. The above argument provides a good kinetic expla- nation of why solid phase epitaxy, with the coherent interface being reactive, usually cannot occur at relatively low tempera-


tures. The argument can also explain why small lattice constant misfit is only a necessary condition for epitaxial growth and not a sufficient condition. This is because in some crystal orientations, although the misfit is small enough so that strain energy due to lattice distortion is small, the activation energy for adjusting the bond structure is rather large, which may prevent epitaxial growth from occurring.

The FEDRs due to the formation process have been discussed in the previous section. Case 2, i.e. Fj, for metastable phase formation larger than F,, for stable phase growth. requires more attention. Although F,,AG,, < F,2AG,2 indicates that the formation of phase 1 will lead to a larger FEDR than the formation of phase 2, it is not the largest FEDR. If the formation process occurs according to the proposed largest FEDR criterion, theoretically the FEDR for this process should be

AC,, + FI;z.AG,z,

where Y” is shown in Fig. IO, and (r&z,) = F,, is the intersection of the vertical dashed line, representing rO, with the formation rate curve of phase 1. It can be reasoned that equation (75) is valid because each molecule transformed from the R region into phase 2 will result in more free energy decrease than that into phase 1. But the number of “molecules” that can be transformed into phase 2 per unit time and unit area is limited and is only equal to F,z,, As a result, the other “molecules” will form phase 1 at a rate of { (rJn,) - Fiz,}. I f phase 3 in Fig. IO also has a larger negative free energy (AC,,) than phase 1, an additional term for phase 3 should be added to equation (75) and the first term should be modified correspondingly. It can be inferred, from equation (75), that although phase 1 formation may be overwhelming when F,, is a few orders of magnitude larger than F,> : the simultaneous formation of phase 2 together with phase 1 is still likely. Because of the difference in the formation rate, phase 2 can only exist as small domains in phase 1. With a decrease in release rate, F,, moves closer to E;,,, while the formation rate for phase 2 remains the same. Consequently, more and more domains of phase 2 appear until F,, passes F,: . I f the structure of phase 1 can be preserved for examination by high resolution analytical techniques: these small domains of phase 2 should be observed. In particular, when phase 1 is amorphous, this inference may be confirmed if small crystalline particles can be observed at temperatures far below the crystallization temperature of this material. Experimental evidence of this inference would also verify the proposed criterion.

The last part of the discussion is devoted to practical concerns regarding the application of the criterion to real systems with significant grain boundary diffusion. strain and different crystalline orientations (Si). In general, grain boundary diffusion is faster than bulk diffusion in the same material. Whether grain boundary diffusion has an effect on silicide formation sequence of a given metal-Si system can be predicted by using RPPs for the system. For example, assume that AJ,, in Fig. 4 represents a bulk diffusion flux in a coarse-grained metal film of a metal- Si couple. In the same metal film, with a finer structure, grain boundary diffusion will dominate. If the grain boundary diffusion flux is so high that it is between JIUc and J,L,c, according to the FEDR criterion, then the first silicides that form in the two samples will be different. Silicide 1 forms in the fine

grain sample, while silicide 2 grows in the coarse grain samples. If the grain boundary diffusion dominated flux is higher than AJ, but not larger than J,LC.r the criterion will predict that silicide 2 forms in both samples.

Elastic strain in the growing silicide will either accelerate or decelerate moving reactant diffusion through the silicide, depending on whether a tensile or a compressive stress exists. How enhanced or restricted diffusion will act on the release rate, and hence silicide formation sequences, can be analyzed and predicted using the same method as discussed in the last paragraph. An elastic strain field in the growing silicide will also affect the formation sequence of the different phases of a given silicide, which can be analyzed and predicted using equation (55), where the elastic strain energy term, AHd, has been included.

Crystalline orientation of the non-moving reactant phase, e.g. Si substrate in a near-noble metalLSi couple, may also have an effect on silicide formation sequence, as discussed in the Introduction. This can be explained by the difference in the chemical bond strength along different crystalline directions. Mathematically, this can be expressed by the magnitude of E. the activation energy for breaking chemical bonds [equations (40) and (59)]. I f the difference in the E values for two orientations is known, one can predict the effect of the orientation on formation sequence using equations (40) and (59) and the RPPs for a given system.

From the discussion above, it can be seen that the silicide formation processes in real systems, with significant GB diffusion, high strain energy and different crystalline orientation, can be analyzed using our recent model and the largest FEDR criterion. as long as the required thermodynamic and kinetic data are available. This demonstrates the advantages of the model and the criterion over others.

CONCLUSIONS

From basic principles of the thermodynamics of irreversible processes, it has been shown that the free energy change due to silicide formation in metaLSi diffusion couples can be determined by examining the free energy change at the reactive interface (or reaction region) located between the growing sili- tide and the non-moving reactant phase. The free energy degradation rate (FEDR) in a reaction region can be expressed as a sum of three contributions, i.e moving reactant diffusion, non- moving reactant release and product molecule crystallization or growth. The interrelationship between these parameters can be demonstrated through the use of a reaction process plot. A criterion for solid-state reactions in metal-Si couples has been derived. Of the number of possible reactions that can occur in a reaction region of a diffusion couple, the reactions producing the largest free energy degradation rate (FEDR) will be those that actually take place. The reactions can be divided into release and formation processes. For the release process, at a given diffusion flux, the release process with the largest release rate will occur. For the subsequent formation process, phase formation resulting in the largest free energy decrease. for the particular silicide generated in the release process, will occur.

Ackno,~l~,d~em~rlts-This work was supported by grants from the Natu-


t-al Science and Engineering Research Council (NSERC) of Canada and the Alberta Microelectronic Centre (AMC).

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Criterion for silicon formation in transition metal-silicon diffusion couples

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