Phase diagrams of thin films and wires

~ui~ri~is Chemistry and Physics, 14 (1986)

57.-68

PHASE DIAGRAMS OF THIN FILMS AND WIRES

M. WAUTELET, L.D. LAUDE and C. ANTONIADIS Institute for Research in Interface Science, FacultC des Sciences, Universite de lIEtat, B-7000 Mons (Belgium)

Received 23 March 1985; accepted 22 April 1985

ABSTRACT Starting from a relation between phonons, defects and surfaces, a model is developped, dealing with the stability of crystalline phases. It is shown that: 1) the disordering temperature decreases with decreasing thickness ; 2) phase diagrams of thin films might differ from bulk ones; 3) surface and interface contributions might modify the previous effects.

INTRODUCTION It is known experimentally that some of the compound phases currently obtained in bulk forms may never appear in thin films. For instance, in silicides, it is often observed that only one of the possible MxSiz phases is actually synthesized in processed thin films [l]. Also, during the preliminary stages of oxidation of metals, one sometimes observes that a given phase appears at first and disappears at a critical thickness above which an other compound develops [2]. Let us also mention the case of laser-induced synthesis of semiconducting films, where only one phase is detected, although the phase diagram presents more than one possibility [3,4]. Crystal surfaces may be considered as a limiting case of thin films and it is well established that surface phase transitions may be different in nature from the bulk ones [5]. Surface atomic configurationsmay not be simply derived from bulk configurations: for instance, the chain-like structure of Si(lll)-(2x1) surface is derived from an assembly of five-and seven-fold rings, while only six-fold ones are present in the bulk 161. Surface segregation in alloys could be another example of the behaviour of surface versus bulk, 0254.0584/86/$3.50

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The fact that the observed thin film behaviour differs from the one expected from the normal phase diagram is often described via the existence of unequal interfacial reaction barriers in the growth kinetics of the multiphase layer structure [7]. The consequence would be that an initial compound phase is favored up to a critical thickness before a second compound phase forms and grows simultaneously.Another proposed mechanism is that there exists a nucleation barrier against the formation of all phases except one [8]. It is also possible that the formation of the missing phases may be thermodynamically impossible. This point of view is taken here. It differs from the two former propositions by the fact that it relates to the equilibrium phase properties while the others relate to the kinetics of phase formations. An indication that our hypothesis is valid in some cases is that very thin films are known to melt at a lower temperature than the corresponding bulk materials [9] and that small aggregates of silicon are amorphous rather than crystalline [lo]. In parallel and because of the present technical tendency to miniaturize electric components, it may be desirable to engineer very thin wires (some 100 i) in a not too far future. It is then also interesting to examine theoretically if bulk phase diagrams would apply to these thin-wired materials. Many aspects of phase diagrams are of interest. Here we choose to concentrate on the melting temperature of different compound phases. In particular, we investigate the possible variation of the melting temperature of different phases upon varying the material thickness. For that purpose, a reasoning is developed from a model recently put forward by the authors in order to explain the origin of disordering of covalent materials [ll] and of their surfaces [12]. It is based on the assumption of a linear relation between the mean phonon frequency of the system and the number of defects.

THE MODEL The total free energy of a crystal is the sum of three contributions : atomic (Fl), phonons (F2) and electronic (F3) [13,14]

F = Fl+ F2+ F3,

Fl = D Ef - kT ln[(N + D)!/N!D!],

(2)

F2 = E, t 3N kT In (Nw/kT),

(3)

where N and D are the total numbers of atoms and intrinsic defects, respectively. G, and Ef are the interaction energy of atoms and the formation energy of defects, respectively.w is the mean phonon frequency. Some authors [15]previously assumed that F3 may be responsible for melting of covalent materials.

59 However, a careful comparison of this theory with experimental data shows that this might not be of importance (16,171 and will not be taken into account in the following. Defects may perturb the vibration spectrum in two ways : either locally and or in a collective manner. In the first case, defects modify F2 via a perturbation of the vibration spectrum around the defects sites, i.e. (with n = D/N)

F2 = E, t 3N (1 - nV)kT ln(tiw/kT)f 3NnV kT In ($w'/kT),

(4)

where V is a measureof the defect perturbed volume of the material, and w' is a local vibration frequency. This contribution allows us to understand the origin of the entropy of formation of defects, Sf. It requires however quantum mechanical calculations for its evaluation. In particular, the description of the high value of Sf (10 - 12 k) in Si and Ge would require either : 1) the existence of extended self-intertitialsor vacancies !18], obtained by replacing every set of 10 atansby llor 9atans inadisordered configuration, or 2) a strong network relaxation around the vacancies, leading to a softening of bonds and a drastic increase of Sf 1191 * But defects may also perturb the delocalized phonon spectrum via the relation

w =wO

(1 - on),

(5)

where w0 and CLare constant [Ill. This relation is easily derived for covalent materials. Indeed in silicon, it has been demonstrated theoretically [20-221 that the phonon spectrum is modified by the creation of electron-hole pairs. This is due to the fact that the transverse acoustic (TA) phonon modes have energies proportional to the electronic charges in the bonds. Therefore the excitation of electron-hole pairs leads to a decrease of w

TA

= win (1 -a'c)

wTA

via

(6)

where c is the concentration of the electron-hole pairs. When vacancies are created in silicon, a similar depopulation of the bonds takes place, since bonding states are transformed into non-bonding ones (the so-called dangling bonds) and

wTA

= tiiA (1 - ciln)

(7)

The other phonon modes (LA, TO, LO) are practically not affected by the population of the bonds [22f. Since the mean frequency entering F2 is defined as the

60 mean value of the logarithm of the frequencies and 25 % of all modes are in the TA branch, while 75 % are in the others, one obtains easily that w iiw0 (1 - cr1n)1'41 W, (1 - oln/4)

(7.a.)

i.e. equation (5) with a = @l/4. The equilibrium concentration of intrinsic defects, n, at a temperature T is obtained in the usual way by zeroing the partial derivative of F relative to n. obtains Introducing eqns. (2), (3) and (5) into (1) and dif&rentiating,one

n = exp (3ol(l

on)) exp (-Ef/kT)

n 3x1-2 2x10 -2 10 ;

D

1600

1700 T(K)

Fig. 1. Values of the density of defects, n, at the extrema of the free energy at different temperatures.The lower part corresponds to the equilibrium values. The curve corresponds to silicon (Ef = 2.35 eV, 01=3.8).

The (n,T) curve is plotted in Fig. 1 for a given set of (Ef,o) values. It appears that, below a critical temperature, Tc, there exist two solutions for n, while there are none above T

This reflects the fact that F(n,T) presents one c' minimum and one maximum below Tc, as shown schematically on Fig. 2. Above Tc, there is no equilibrium value of n and the crystal is unstable. In the case-of systems possessing only one ordered phase, Tc may be associated with the melting temperature Tm, at which aT/& = 0, i.e,

a?$! - (30r2+ 2oI)nc+ 1 = 0

(9)

61

Fig. 2. Schematic representation of the free energy below and above Tc (see text)

Then for each ~1,there is one corresponding value of nc and one value of (kTm/Ef) deduced by putting nc into eqn.(8). In other words, the knowledge of only one of the parameters o, n

C

or (kTm/Ef) allows us to deduce the other

two. Moreover two parameters are, in principle, measurable. This is immediate for (kTm/Ef). Also, 3ok = Sf , the entropy of formation of the corresponding defects, as seen from eqn.(8) when (1 -a n) 2 1. The unique relation between (kTm/Ef) and Sf compares well with the experiments for a variety of materials [11]. In principle, eqn.(4) has also to be taken into account. When it is introduced in the problem, it gives rise to an increase of Sf, but the instability remains due to the -delocalized contribution of defects to the phonon spectrum. In crystals, intrinsic defects are taken to be equivalent to internal surfaces In other words, the role of the surface on

w=~o

(l-

W

is similar to that of defects and

au-an)

(10)

where ois a parameter proportional to the surface to volume ratio and

to the

surface to defect ratio. For instance, in silicon, o= 1 correspondsto 4 dangling bonds, since one defect (monovacancy) possesses four dangling bonds. Including eqn.(lO) into F, one derives

n = exp [ 3d(l - ao-on)]

c?n* -ant C

(2 - 2oo+ 39

exp (-Ef/kT)

(11)

+ (1 -uU)2 = 0

(12)

62 (1 -ou - url)

kTm _ 5--3a

-

(13)

(1 -au - ant) ln fn,)

From these equations, one deduces immediately that, at a given a, and increasing o, Tm decreases (Fig.3.) while nc decreases. Also, Tm decreases more rapidly with increasing o when o is larger.

-@/TKl

1”

-____---_

-----

0.5-

1 50

100

w v&f

Fig.3. Variation of the disordering temperature, T , with the ratio of numbers of surface to volume atoms, for different values oFa (see text),

DISORDERING OF THIN FILMS AND WIRES From the previous reasoning, disordering may be associated with an instabim

of the concentrationof intrinsic defects. This is a catastrophic effect

with a sharp transition. Moreover, the 'order parameter' of the system is a measurable quantity, namely the concentration of defects. This has to be compared with some of the numerous theories of melting, more precisely those which relate melting to a catastrophic proliferation of holes [23] or dislocations [%25],

or which associate phonons and defects [26]. The present theory seems

more compatible with the hole theory than with the dislocation theory of melting. However, it has to be borne in mind that it may explain the origin of disordering from parameters relevant to the solid state alone, but carries no information on the kinetics of the melting process : how defects proliferate, how the order-disordertransition occurs (via defects, or symmetry related properties

63

[27] or anharmonicity [28] or disclinations 1291, ...). i.e. the present theory says only that the crystal is unstable at Tm . Also, it does not imply that melting is a second order phase transition, since it does not describe the system up to the molten state. As said before, the model predicts a decrease of Tm with increasing o, i.e. with decreasing thickness or diameter. Let us recall that Tm does not necessarily refer to melting but also to disordering (amorphsation), since melting implies an atomic (or vacancy) mobility, not necessarilypresent at low temperature. This may explain the experimental fact that metallic films deposited on amorphous substrates are disordered when the deposition temperature exceeds roughly (2Tm/3) but are crystalline below that temperature [9]. In order to describe this peculiarity, let us look at Fig. 3. At a temperature To, films with o larger (thickness d small) than a value given by the o(To) curve are not crystalline, while they are (d larger) if o is less than this value. When To increases, the corresponding o decreases (d increases). Just below Tm , 0 goes to zero (dgoes to infinity). A precise comparison of experimental data with the theory is unfortunately difficult due to two other contributions : 1) very thin films appear often as assemblies of globules, i.e. with a larger 0 than the flat films ; 2) well below Tm, defects due to the deposition itself are not very mobile and contributealso to u

191.

Electrical or EPR data would perhaps permit to measure this last contribution. Qualitatively, the model permits also to understand the fact that films [30] may appear first as disordered globules and crystallize suddenly when they agglomerate. This would be due to the larger o of globules as compared to homogeneous thin films, resulting in a lower disordering temperature of globules. The case of thin wires is similar but the reduction of Tm with decreasing dimension is more dramatic, due to the larger number of surface to volume atomic ratio. The case is also complicated by the fact that the surface of thin wires is not flat and presents probably more than one crystalline face, i.e. more than one oeff. Whatever the exact shape of the wire, it is obvious that it may disorder at low temperature. An order of magnitude estimate of the very minimum size below which the crystal will never be stable is given by not = 1, since kTm may never attain 0 K. For silicon, given cr= 3.8, one obtains 0

C

kc/2

= 0.26 and a relative number of surface atoms between 4oc/3 = 0.35 and =

0.52 (since one bulk vacancy contains four dangling bonds). Taking a

mean value of 0.4, the critical diameter of awire would attain 10 interatomic distances, i.e. about 20 A , and contain 75 atoms per layer. The corresponding critical diameter of a cluster would attain 30 A and contain about 1700 atoms, in good agreement with other data [lo]. This point would have to be taken into account in any investigation of the properties of very thin wires.

64

PHASE DIAGRAM OF THIN FILMS As said in the introduction, the phases observed in bulk materials may be absent in thin films. For instance, only few MxSi compounds are observed. In Y the framework of our model, the interesting point is that different compositionalphases are characterised by different intrinsic defects and, hence, different defect-phononcoupling, a.. Let us consider two phases, AxBy and A B ,

xz

1

characterised by different melting temperatures of bulk phases, Ti and T;, and different

s : a1 and a2. Let l/u be proportional to the surface area. Two i kinds of situations exist, depending on the (Ti/Ts) and (~/a~) ratios, as shown a

schematicallyon Fig. 4. When Ti>

Ts and "15 02, the AxBz phase becomes always

unstable at a temperature lower than the AxBy phase, so that the bulk phase diagram remains qualitativelyvalid. When TT > T;

and a1 > c$, a crossing of the

Below the corresponding cricr' tical thickness, the bulk phase diagram does not apply. Indeed, at low temperainstability temperature occurs for a criticalo

ture, both crystalline phases are present within their own composition range, while in a certain temperature range, the 'high temperature phase', (AxBy), is melted (in the thin film regime), while the other one (AxBz) remains crystaof lline. In order to apply the reasoning to one particular case, a knowledge. the properties of intrinsic defects and of the defect-phonon coupling are obviously required but are unfortunately not available. The previous discussion may also serve as a basis for the understanding of the occurence of metastable phases in the thin film regime. For instance, bulk

a

b

Fig.4. Schematic diagrams of the variation of theldisorderingtemperatures of two phases, when the thickness (proportionalto o ) varies. a) TT > T; ; al< a2 (see text) ; b) Ti > Tz ; al > a*.

65 CdTe or CdSe has a stable cubic structure and a metastable hexagonal one [9]. But an excess of the metal atoms increases the stability of the hexagonal phase in the films. Within the present theory, an excess of one kind of atoms means an increase of the number of intrinsic defects by u (eqn.(lO)). This leads to a decrease of the stability temperature of the phase, similarly to the amorphisation of very defective materials [ll]. If the other 'metastable'phase reacts differently (i.e. CY is less than for the stable phase), the stable phase may disorder at a lower temperature than the metastable one does. At this stage, the proposed mechanism is obviously speculative, but may be checked experimentally by evaluation disorder as a function of a departure from stoichiometry.

OTHER EFFECTS In the previous discussion, it is assumed that the surface is characterized by a given value of (CIU).However, depending on the experimental conditions, it may well be that (ao)is modified. At least two effects may contribute to a variation of(ao)

: surface reconstruction and chemisorption.Let us look

at Si(ll1) as an example.

Surface reconstruction Si(ll1) is known to exhibit two structures at room temperature and in ultrahigh vacuum. Just after cleavage, a metastable (2.x1) structure is observed. It transforms irreversibly into the (7x 7) structure on heating for some time at moderate temperature . This has been interpreted in the framework of the present theory (in two dimensions) [12], as due to a variation of (oo)surf between the two structures. The number of dangling bonds being larger on the (2 x 1) than on the (7 x7) surface, the corresponding a~(2 x 1)s c1~ (7 x 7). In other words, equatin (13) indicates that thin (lll)-oriented Si films with (2 X 1) surface would disorder at a lower temperature than the (7x7) one. It would be interesting to test this point experimentally.

Chemisorption Let us consider oxidation of a Si(ll1) surface. Since chemisorption involves a modification of the bond charges, the strong electron-phonon coupling [12] occuring at the surface modifies the value of CI for surface sites, i.e. the term (a~). Since oxygen atoms have tendency to capture electrons, one expects a transfer of electrons from the Si orbitals to oxygen ones. In other words, the Si states are less bonding than a free surface, ac1(02)

>a(clean), and one expects a decrease of the disordering temperature

of the film upon oxygen chemisorption.

Interfaces The deacrease of Tm with increasing ~1also implies that thin 'interface'films may disorder at a lower temperature than the bulk material. For instance, let us consider two elements, A and B, in contact. They are further assumed to react, i.e., at the interface a film of the compound say AB, is formed. One may also consider thin films, some 100 i thick (or less), of one material inserted between two other materials. These thin films are characterized by : i) a large o due to the interface layers ; ii) a value of CL (for the surface) which differs from the free surface one (similarlyto the case of chemisorption) ; iii) a second contribution to u , due to mismatch within the atomic network. This leads obviously to a decrease of Tm of the interface. This has important consequences in a number of applications. For instance, let us consider the case of laser-induced synthesis of semiconductingfilms, starting from the elemental constituents [31]. It

appears that the threshold

energy of formation of the compound (say,AlSb) increases drastically when the starting (sandwich-like)Al and Sb films are pre-annealed at high temperature. Formation of thick AlSb grain boundaries is expected under these conditions. Since : i) the reaction is believed to occur in the molten state [32]; ii) the bulk compound AlSb melts at a much higher temperature than the elemental constituents, it may be argued that the thinner AlSb intermediate layers (present between Al and Sb grains under low temperature pre-annealing)melt at a lower temperature than the thicker ones ( present under high temperature annealing). The same effect is expected to be of primary importance under any energy pulse treatment of interfaces in which melting of the constituentshas to be achieved before a 'thick' interface develops. Such an interface would develop under slow annealing conditions and would inhibit the 'liquid state mixing' of the reactants.

Debye temperature The present model would imply that the Debye temperature, s, of very thin films, wires and microcrystalsmay be less than that of the bulk. Indeed, in a first approximation,kOD = hwo(l-ao

) . In principle this could be seen in

X-ray diffraction, via the Debye-Waller factor. Unfortunately,other effects contribute to the line broadening by small particles [33]. The same would be true for the Lamb-MGssbauerbroadening 1341. Indications that this statement is correct come from a recent work [35] in which C+,deduced from resistivity measurementsis smaller for thin Au films than for the bulk material. Further work is obviously required to solve this point.

67

Since the specific heat is related to CD, the present theory indicates that it is higher for thin films than for bulk materials ; a fact that has to be taken into account mainly for materials with bulk $ much higher than room temperature.

CONCLUSIONS Altogether, it appears that the present model permits an understanding of the origin of the phases observed in thin films synthesis, on the basis of phonons-defects-surfacesinteractions.

ACKNOWLEDGEMENTS The authors thank Drs R. Andrew, L. Baufay, P. Viscor for fruitful discussions.

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