J. Phys. D: Appl. Phys. 30 (1997) 3167–3186. Printed in the UK PII: S0022-3727(97)69955-6

REVIEW ARTICLE

Investigating the thermodynamicsand kinetics of thin film reactions bydifferential scanning calorimetry

C Michaelsen †, K Barmak ‡ and T P Weihs §

† Institute of Materials Research, GKSS Research Centre, 21502 Geesthacht,Germany‡ Department of Materials Science and Engineering, Lehigh University, Bethlehem,Pennsylvania 18015, USA§ Department of Materials Science and Engineering, Johns Hopkins University,Baltimore, Maryland 21218, USA

Received 15 July 1997

Abstract. In this paper we demonstrate the utility of differential scanningcalorimetry for investigating the thermodynamics and kinetics of a broad range ofthin film reactions. We begin by describing differential scanning calorimeters andthe preparation of thin film samples. We then cite a number of examples thatillustrate how enthalpies of crystallization, heats of formation and enthalpies ofinterfaces can be measured using layered thin films of Ni/Al, Cu/Zr and Zr/Al andhomogeneous thin films of Co–Si, Nb–Cu, Cr–Cu and Ge–Sn. Following theseexamples of thermodynamic measurements, we show how kinetic parameters ofnucleation, growth and coarsening can also be determined from differentialscanning calorimetry traces using layered thin films of Ni/Al, Ti/Al and Nb/Al andhomogenous thin films of Co–Si and Ge–Sn. The thermodynamic and kineticinvestigations highlighted in these examples demonstrate that one can characterizephase transformations that are relevant to commercial applications and scientificstudies both of thin films and of bulk materials.

Definitions of symbols

β heating rateC heat capacityc,1c composition, composition changeH,1H enthalpy, enthalpy changeE activation energyg, g∗ geometrical factors for the interfacial enthalpyγH enthalpic part of the interfacial tensionK heat transfer coefficientk rate constantk0 pre-exponential factor of the rate constantkB Boltzmann’s constant3 multilayer period lengthm sample massM molecular weightn Avrami exponentN nucleation site densitynmole number of atoms in one moleculep grain growth exponentr radiust timeτ time constant

T temperaturev growth velocityV molar volumew half thickness of an intermixed layerx thicknessX transformed volume fraction

1. Introduction

Thin films play an important role in the advance of manymodern technologies such as electronic devices, storagemedia, read/write heads, flat panel displays, chemicalsensors, protective coatings and x-ray optics [1–3]. Sucha remarkable array of applications creates a sense ofexcitement among thin film scientists and engineers thatis bound to continue into the 21st century. This excitementis further fuelled by the development of new applications.For example, engineering the interfacial region in advancedcomposite materials using multilayer thin films [4] andjoining components using reactive multilayer foils as localheat sources [5–7] are two promising applications withenormous potential. The development of other applications

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is expected as improvements in processing techniques allownovel thin film structures to be made cost effectively.

In all existing or potential uses of thin films, thepromotion or inhibition of reactions, such as compoundformation, crystallization and grain growth, plays a centralrole in the ultimate performance of the system. Whetherthe former or the latter is desired, it becomes necessaryto understand the thermodynamic and kinetic details ofthe reaction. The need for this understanding continuesto drive basic research in this area. However, thisresearch is not limited to those systems with technologicalapplications. Thin films also provide model systems forfundamental studies of a number of reactions that are noteasily accessible in bulk systems. These studies contributesignificantly to our understanding of materials’ behaviourand create stimulating new opportunities for scientificinvestigations. They also provide a means for the designand optimization of materials. Thus we see three areas thatbenefit tremendously from research on thin film reactions:thin film technologies, materials engineering and materialsscience.

A critical step towards understanding thin filmreactions is to characterize their thermodynamic and kineticparameters. Differential scanning calorimetry (DSC) hasproven to be a most promising technique in this effort andthe present review gives various examples of its utility.Most of the examples have been chosen from our ownstudies primarily because of our deeper familiarity withthem. However, for each example we reference a variety ofworks by other researchers on the same topics. In fact, wewish to make a special mention of an earlier review articleby Spaepen and Thompson [8] on calorimetric studiesof reactions in thin films and multilayers. Their reviewdiffers from the present article in its approach. Spaepenand Thompson chose the material system and then showedhow DSC could be used to determine the thermodynamicor kinetic parameters of interest. In contrast, we choosethe parameters and then select the material system todemonstrate their measurement. The two works aretherefore highly complementary.

This paper is organized as follows. Section 2 gives therudiments of DSC instrumentation and presents in generalterms the techniques for sample preparation. Section 3describes how enthalpies of crystallization, formation andinterfaces can be measured using DSC and homogeneousor layered thin films. Section 4 demonstrates how kineticparameters of nucleation, growth and coarsening can alsobe determined from the DSC traces. The goal in sections 3and 4 is to show that, by making a significant effort in thedesign and fabrication of novel thin film samples, one canmeasure many thermodynamic and kinetic parameters ofinterest. Section 5 provides a short summary of the paper.

2. Experimental procedures for DSC studies

2.1. The instrumentation

When heated to high enough temperatures, all materialsundergo physical or chemical changes whether or not theyare bulk materials or thin films. These changes alter the

enthalpy and/or heat capacity of the material which inturn results in the release or the absorption of heat. Bydetermining the instantaneous rate of heat flow (namely,the power), differential scanning calorimetry providesquantitative thermodynamic and kinetic information aboutthe physical and chemical changes occurring in the material.

The terms ‘differential scanning calorimetry’ and‘differential scanning calorimeter’ and their abbreviation,‘DSC’, were first introduced by Perkin Elmer Corporationto distinguish the Perkin Elmer system from otherexisting instruments for differential thermal analysis. Thisdistinction has been described in detail elsewhere [9–13]and here we give only a brief summary. In the PerkinElmer DSC the sample and the reference are thermallyinsulated from one another and each is provided withits own individual heater, as shown schematically infigure 1(a). The instrument is based on the ‘null-balance’principle of measurement in which the energy absorbedor evolved by the sample is compensated by adding orsubtracting an equivalent amount of electrical energy tothe heater located in the sample holder. In practice,this is achieved by comparing the signal from a platinumresistance thermometer in the sample holder with thatfrom an identical sensor in the reference holder. (Thesethermometers need to be calibrated, since the true sampletemperature will deviate from the sensor temperature as aresult of thermal lags that become increasingly pronouncedat high heating rates [14, 15].) The continuous andautomatic adjustment of the heater power necessary tokeep the sample holder temperature identical to that ofthe reference holder, that is to keep1T = 0, providesa varying electrical signaloppositebut equivalent to thepower absorbed or released by the sample. Thus, the PerkinElmer system is apower-compensatedDSC.

Unfortunately, over the years, the distinction betweenthe power-compensated DSC and other differential thermalinstruments has been lost, resulting in what is certainto continue to be a state of confusion with regard tonomenclature in differential scanning calorimetry [9, 10].Two differential thermal systems that are frequentlyconfused with the power-compensated DSC are (1) the‘classical’ differential thermal analysis (DTA) system and(2) the ‘Boersma’ DTA which was named after the authorof the pioneering paper [16]. In both DTAs only asingle heating source is used and the sample and referencecrucibles are arranged symmetrically with respect to thisheating source. In the classical DTA, temperature sensorsare located in the interior both of the sample and ofreference materials and they record an uncompensatedtemperature difference,1T , between the sample and thereference material (figure 1(b)). In the Boersma DTAtemperature sensors are located on the exteriors of thesample and reference (or their containers) and again areused to record an uncompensated temperature difference(figure 1(c)) [9, 10].

Whereas the power-compensated DSC measures thepower signal directly, a signal-to-power conversion in theclassical and Boersma DTAs requires the determinationof the two thermal constantsK and C in the equationpower= K1T + C d1T/dt , whereK is the heat transfer

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Figure 1. Schematic representations of three differentialthermal analysis systems: (a) DSC, (b) classical DTA and(c) Boersma DTA.

coefficient andC is the heat capacity [17]. The derivationof this equation involves several assumptions that werediscussed by Chen and Kirsch [17]. The determinationof the two thermal constants requires a large series ofcalibration runs using materials with known transformationtemperatures and enthalpies [18]. Once the raw1T datahave been converted, the rate of enthalpy change providesa calorimetric measurement. Thus, it is easy to understandthe origin of the confusion in the use of the generic termDSC. The classical and Boersma DTAs have been termedheat-fluxDSCs in cases in which care has been taken tomake the distinction [10, 18, 19].

The DSC’s ability to measure the power of a reactiondirectly, in addition to the DSC’s higher sensitivity andshorter time constant, is extremely beneficial in the studyof thin film reactions. Thus, many investigators favourthis type of DSC. However, radiative heat losses becomesignificant at higher temperatures and these heat lossesare in conflict with the measurement principle of power-compensated DSCs. As a result, DSCs have a lowermaximum operating temperature than do DTAs and somethin film reactions cannot be completed in a DSC. Studies ofthese reactions therefore require the use of heat flux DSCswhich have significantly higher operating temperatures. Inthe following sections we provide examples of the use of

Figure 2. A transmission electron micrograph of the crosssection of a 1Nb/3Al multilayer film with 3 = 72 nmdeposited on a sapphire substrate. The substrate can beseen at the bottom of the micrograph.

both calorimeters. Lastly it is worth noting that special thinfilm calorimeters have been designed for samples with verysmall masses. A detailed description of such an instrumentwas given in [20].

2.2. Sample preparation

Thin film deposition methods are indeed numerous [21] andnew methods are continuously being developed [1]. Thethin films described in the present review were all depositedby magnetron sputtering or electron beam evaporation,two well established techniques for producing high-qualityinorganic films. Depending on the sensitivity of theinstrument and the nature of the reaction, the sample massnecessary to obtain good signal-to-noise ratios ranges froma fraction of a milligram to approximately 20 mg. Inorder to increase the signal-to-noise ratio further, threetechniques can be employed. First, when interface reactionsare investigated, the film can be fabricated as a layeredstructure with many reacting interfaces that increase therate of reaction for a given mass, thereby amplifying thesignal. An example of such a multilayered film is shown infigure 2. Second, the film can be removed from its substrateand tested as a free-standing sample. This maximizes thesignal for a given mass of sample in the DSC. Third, scanscan be performed at higher heating rates which increasethe power signal relative to the baseline noise. However,care must be taken to avoid loss of null-balance power-compensation at higher heating rates.

In order to generate free-standing thin film samples,two techniques have been employed. In one case, thefilm thickness was increased to greater than 10µm sothe film could be removed from the substrate without

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Figure 3. A typical DSC scan yields a measure of the rateat which heat flows from or into a sample as it reacts. Theschematic diagram for the first scan shows two exothermicreactions. If these reactions are irreversible, a moreprecise measure of the heat flow can be obtained byscanning (heating) the sample a second time to obtain abaseline. The difference in heat flow between the first andsecond scans can then be integrated to obtain the totalheat of each reaction. Here, we have integrated only partway through reaction 1.

tearing or curling [22, 23]. This removal was facilitatedby choosing an initial layer with limited adhesion to thesubstrate. In the other case, a soluble underlayer was puton the substrate before the film of interest was deposited.The choice of the soluble layer depends on the materialsystem and the deposition method and may require lengthyexperimentation, because it is necessary to dissolve thelayer completely without attacking the film. In manyinstances this underlayer is a photoresist [24], but in otherinstances it may be a metallic layer [25–27].

Once a free-standing film has been obtained, it ispacked into a metallic or ceramic holder that is suitablefor the temperature range and sample of study. An emptyholder of nearly identical mass to the sample holder plussample is put into the reference furnace or crucible. Thecalorimetry experiments are then carried out either at aconstant heating rate or isothermally. In both cases itis extremely important to have obtained a flat baselineprior to running any experiments, particularly for reversibletransformations. For irreversible transformations a moreaccurate baseline can be obtained in a second heating cycleafter cooling the reacted sample to the starting conditionswithout disturbing it and then reheating (see figure 3). Ofcourse, it is important that the reaction will have gone tocompletion during the first heating cycle.

A comprehensive study of reactions in thin filmsshould also utilize other characterization techniques thatcomplement differential scanning calorimetry. Forexample, product phases associated with each thermalsignal are often identified using x-ray or electron diffractionmethods. Furthermore, the unambiguous interpretationof kinetic data may require the characterization of afilm’s microstructure. Since thin film microstructuresoften have nanometre-to-micrometre length scales, thischaracterization frequently demands the use of transmission

electron microscopy. These points will be clarified throughthe examples given in the following sections.

3. Thermodynamics

The enthalpy released or absorbed during a phasetransformation is an important thermodynamic quantity thatis commonly used to predict the stability of alloys andcompounds [28–30]. These enthalpies can be determinedfrom DSC experiments by integrating the rate of heatflow over the course of the reaction. This integrationis schematically demonstrated in figure 3 for a constant-heating-rate experiment and takes the form

1H =∫ T2

T1

1

β[(dH/dt)dat − (dH/dt)bsl ] dT (1)

where1H is the total heat released,β = dT/dt is theheating rate,(dH/dt)dat is the sample heat flow rate,(dH/dt)bsl is the baseline heat flow rate andT1 and T2

are the limits of integration. To calculate the enthalpyper mole of atoms of the product phase (equivalent tog-atom), the integral1H must be multiplied by the constantM/(mnmole), where M is the molecular weight of theproduct phase,m is the mass of the sample andnmole is thenumber of atoms in one molecule of the compound phase.This enthalpy is typically referenced to a temperaturethat marks the onset of the reaction. However, one canreference the enthalpy to room temperature by choosingT1

appropriately [31].Using the above procedure we describe how one

can measure enthalpies of formation, enthalpies ofcrystallization and enthalpies of interfaces in the followingsubsections. We discuss the formation of three stable(equilibrium) compounds, three metastable phases and oneunstable alloy. In addition, we will consider both intraphaseand interphase interfaces. Our goal is to show that, byfabricating appropriate thin film samples, one can measurethe enthalpy changes for a broad range of transformationsat relatively low temperatures using differential scanningcalorimetry.

3.1. The formation of equilibrium compounds: NiAl3,Cu51Zr 14 and ZrAl x

Most enthalpies of formation have been measured usinghigh-temperature calorimetry techniques such as solute–solvent drop calorimetry and direct synthesis calorimetry[29, 30, 32–34]. In describing these methods, Kleppa [29]and Kubaschewski and Alcock [30] noted that radiative heatlosses and environmental contamination are major concernsbecause measurements are often performed at temperaturesabove 1300 K. These two potentially significant concernscan be minimized or eliminated with the combination ofthin film samples and differential scanning calorimetrydescribed here. The fine microstructures of the filmsprovide short diffusion distances that greatly enhanceatomic mixing compared with that in bulk materials.As a consequence, many formation reactions (and manyother reactions) can be completed rapidly in a DSC at

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Figure 4. DSC traces of 1Ni/3Al multilayer films withvarious periods, 3, measured at a heating rate of40 K min−1 [27].

temperatures below 1000 K. At these lower temperatures,radiation and contamination are significantly reduced andenthalpies of compound formation can be measured veryprecisely (<±2.0%) [31].

As a first example of using thin film samples to measureenthalpies of formation of equilibrium compounds, considerthe DSC traces in figure 4. These data were obtained for theformation of NiAl3 in Ni/Al multilayers with a fixed overallstoichiometry of 1Ni:3Al [27]. By integrating the heat flowwith respect to time, the heat of reaction was obtained foreach sample and is plotted versus the multilayer period,3,in figure 5. Figure 5 clearly demonstrates that the heatof reaction dropped as the multilayer period decreased,indicating that some intermixing and reaction occurredduring the deposition. The enthalpy of formation of NiAl3

was estimated to be−36.2 kJ g-atom−1 by averaging theheats of reaction at large periods, which approach a constantvalue. Although this enthalpy of formation (1Hf ) is closeto the literature value of−37.7±2.1 kJ g-atom−1 [35], moreaccurate measures of1Hf can be obtained by utilizingmodels that account for the heat that is lost when thesamples begin to intermix during deposition [7, 31].

The heat lost due to intermixing is approximated bymodelling the composition profile across the intermixedlayer in the as-deposited samples. This profile gives thedegree to which unlike bonds (Ni—Al) have replaced likebonds (Al—Al, Ni—Ni) and therefore estimates the degreeof heat loss. We consider two types of profiles. In the firsttype, the intermixing forms a chemically invariant phasesuch as a line compound with one distinct composition.This results in a sharp chemical profile, as shown in

Figure 5. Heats measured for the transformation of Ni/Almultilayers to NiAl3 [27] are plotted versus the multilayerperiod, 3, together with fits of equations (4) and (5) and aprevious measure of the enthalpy of formation of NiAl3.Note that the DSC heats extrapolate to the centre of thegrey scatter band for the value of −37.7± 2.1 kJ g-atom−1

listed in [35].

figure 6(a). In the second type, intermixing forms achemically variant phase such as a solid solution witha broad composition range. For this case, a smoothcomposition profile results, as shown in figure 6(b). Here,we assume that the composition varies from a value of 1(all Ni) to a value of−1 (all Al). The smooth compositionprofile is then approximated by

c(x) = 1− exp

(− ln 2

wx

)0≤ x ≤ 3

4(2a)

c(x) = −1+ exp

(ln 2

wx

)− 3

4≤ x < 0 (2b)

where c(x) is the composition at a distancex from theinterface and 2w defines the thickness of the intermixedlayer. c(x) is equal to+0.5 atw and−0.5 at−w.

To determine a compound’s enthalpy of formationwhile usingc(x) to account for intermixing, we first assumethat the heat production varies linearly with changes incomposition:1H ∝ 1c. Then we integratec(x) to obtaina relationship between a sample’s heat of reaction,1H ,and the compound’s enthalpy of formation,1Hf . Forthe simplest case of a sample with equal layer thicknessesand a compound with a 50:50 composition, the integrationsimplifies to

1H = 1Hf

3/4

∫ 3/4

0c(x) dx. (3)

For samples with sharp chemical profiles (figure 6(a)), therelation between1H and1Hf is then given by

1H = 1Hf(

1− 4w

3

)(4)

where 2w is the thickness of the chemically distinctintermixed layer. For samples with smooth chemical

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Figure 6. Two possible composition profiles in multilayer samples containing elements A and B that have begun to intermixduring their deposition. In (a) the intermixing leads to the formation of a compound with a distinct composition between the Aand B layers. In this case the composition steps abruptly from all A (c = 1), to the composition of a particular compound(c = 0 for AB), to all B (c = −1). In (b) intermixing leads to the formation of a compositionally variant alloy or compound and asmooth composition profile.

profiles (figure 6(b)), the relation is given by

1H = 1Hf{

1− 4w

3 ln(2)

[1− exp

(−3 ln(2)

4w

)]}.

(5)With either relationship we can determine enthalpies offormation using a series of samples that have differentmultilayer periods, but were deposited under identicalconditions and have similar intermixed zones and averagecompositions. Note that both equations (4) and (5) reduceto the equality1H = 1Hf at infinitely large periods,suggesting that either equation can be used to determine1Hf for a given compound. For the Ni/Al multilayers,both equation (4) and equation (5) were fitted to the1Hdata and the results are replotted versus 1/3 in figure 7.The model which assumes a smooth composition profileclearly produces the best fit to the data, particularly at smallperiods [7, 36]. The enthalpy of formation of NiAl3 wascalculated to be−37.9± 0.5 kJ g-atom−1 by extrapolatingthe fitted curve to an infinitely large multilayer period. Thisvalue matches the earlier high-temperature measurement for1Hf [35] and falls well within the scatter band of theseearlier measurements, as shown in figure 7. The low scatterof the DSC measurement demonstrates that enthalpies offormation can be measured precisely at low temperaturesusing DSC and multilayer samples.

In contrast to the Ni/Al system, one obtains the best fitsto the heats of reactions in the Cu/Zr system by assuming astepped composition profile as shown in figure 6(a) [31]. Inthis example three distinct exotherms were observed whenCu-rich, Cu/Zr multilayers were heated in a DSC, as shownin figure 8. The initial exotherm is due to a solid stateamorphization of the thinner Zr layers; the second exothermis due to the formation of the line compound, Cu51Zr14, andthe third exotherm is due to the formation of the equilibriumphase, Cu9Zr2 [23, 31]. The TEM micrographs that are

Figure 7. Heats measured for the transformation of Ni/Almultilayers to NiAl3 [27] are replotted versus the inversemultilayer period, 1/3, together with fits of equations (4)and (5). Note that all values and fits fall on one line at largeperiods. The intercept of this line has a value of−37.9± 0.5 kJ g-atom−1 which matches well with theenthalpy of formation, −37.7± 2.1 kJ g-atom−1, listed in[35].

schematically reproduced in figures 9(a)–(d) chronicle theprogress of the reactions.

Since the first reaction in figure 8 starts with crystallineCu and crystalline Zr and the second reaction ends withcrystalline Cu and crystalline Cu51Zr14, the combinedheats of these two reactions (1H12) can be used toquantify Cu51Zr14’s enthalpy of formation [31].1H12 wascalculated by integrating the DSC traces in figure 8 whilecorrecting for the unreacted Cu. The resulting values areplotted versus the reciprocal of the Zr layer thickness infigure 10. (The Zr layer thickness was used in place of the

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Figure 8. Net heat flows from DSC scans of five Cu/Zrmultilayers with different compositions. The heating ratewas 100 ◦C min−1 and the heat flows were calculated permole of atoms [23, 31].

Figure 9. A schematic diagram that shows the evolution oflayering and phases in Cu/Zr multilayers. The as-depositedsamples contain alternating layers of polycrystalline Cu andpolycrystalline Zr with some solid state amorphizationbetween the layers. After heating beyond the firstexothermic reaction all of the Zr and some of the Cu isamorphized to yield uniform layers of amorphous Cu–Zrand Cu. The amorphous Cu–Zr then reacts with some ofthe excess Cu to form alternating layers of Cu51Zr14 and Cuin reaction 2. In the final exothermic reaction the uniformlayering breaks down and large grains of ordered Cu9Zr2form and encompass much smaller grains of Cu [23, 31].

multilayer period because the sample contained excess Cu.)The enthalpy of formation of Cu51Zr14 was calculated to be−14.3±0.3 kJ g-atom−1 using the linear fit to the data andequation (4). Note that this intercept value is very closeto the high-temperature drop calorimetry measurement of−14.07± 1.24 kJ g-atom−1 by Kleppa and Wanatabe [32]but the data have less scatter.

A final example of the measurement of1Hf is shownin figure 11 for the Zr–Al binary system [37]. Formationenthalpies were measured for five different compositionsusing multiple DSC traces and samples, and the resultsare compared with bulk values obtained by two differenttechniques. The ‘DSC’ values in figure 11 match theenthalpies measured by Meschel and Kleppa using high-

Figure 10. The integrated heats of reactions 1 and 2 werecalculated per mole of Cu51Zr14 that forms in each sample[31]. As the initial Zr layer thickness increases, the heatsapproach the enthalpy of formation for Cu51Zr14 that wasmeasured by Kleppa and Wanatabe [32]. This value ismarked as a broken line and their scatter is indicated bythe shaded region.

temperature, direct synthesis calorimetry [38] very closely.The only exception is the sample with 46.8 at% Zr. For thisparticular sample, temperatures above the 1000 K limit ofthe DSC were needed to form the equilibrium compounds,AlZr and Al3Zr2, and the actual formation enthalpy ispredicted to be higher on the basis of a separate DTAanalysis [37]. The DSC values in figure 11 also agreewith most of the formation enthalpies that Kematick andFranzen measured using Al vapour pressures [39]. The onlydeviation appeared at high Zr concentrations, for which Alvapour pressure measurements become suspect. Thus, ingeneral, the Al–Zr data in figure 11 demonstrate again thatlow-temperature DSC can be used to determine formationenthalpies for stable intermetallic compounds.

3.2. The formation of metastable alloys: amorphousCu–Zr

In addition to stable phases, differential scanningcalorimetry can be used to investigate the formation ofmetastable phases. These phases often appear in alloythin films synthesized by the co-deposition of variouselements and in multilayer films during low-temperatureanneals. An example of metastable phase formation isthe solid state amorphization that occurs in the Cu/Zrmultilayers described earlier. The first exotherm in eachDSC scan in figure 8 corresponds to the formation of ametastable amorphous phase. The enthalpy of formationfor the amorphous Cu–Zr was determined by deconvolutingthe first exotherm from the second overlapping exothermin figure 8 using two different methods [23]. (Sinceboth methods have merits when deconvoluting exotherms,both are described briefly.) In one method, Cauchydistributions were fitted to the DSC scans (see also [40]).By integrating the first distribution for exotherm 1, the heat

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Figure 11. Enthalpies of formation that were measured forZr–Al alloys using low-temperature DSC scans [37] arecompared with similar values from high-temperaturecalorimetry experiments. Meschel and Kleppa [38]measured enthalpies using direct synthesis calorimetry andAl and Zr powders whereas Kematick and Franzen [39]measured enthalpies of formation using a Knudsen cellmass spectrometric technique and Al vapour pressures.Note that both sets of data agree with the DSC resultsexcept for the Zr-rich samples. The enthalpy of formationfor the 46.8 at% Zr sample, that did not form its equilibriumphase in the DSC scans, is estimated to be 29% higher or−52.1 kJ g-atom−1.

of amorphization was calculated to be1H1 = −2.38 kJper mole of atoms in the sample with 8.1 at. % Zr. In theother method, samples were first annealed isothermally tocomplete the amorphization before performing a constant-heating-rate experiment. After this pre-anneal, the firstreaction was almost entirely absent from the experimentaldata, whereas the second and third reactions had notdiminished. By measuring1H2 and subtracting this valuefrom the combined heat for both exotherms,1H1 wascalculated to be−2.57 kJ per mole of atoms in the samplewith 8.1 at. % Zr, similar to the value from the first method.

The heats of amorphization in the Cu/Zr multilayerswere normalized to account for the composition (34 at%Zr) and volume fraction of amorphous phase that formed[23] and the results are plotted in figure 12. The formationenthalpy of this Cu-rich amorphous phase was calculated tobe−13.8±0.9 kJ g-atom−1 using a linear fit to the data andequation (4). Although the precision of this measurementmay be low due to scatter in the data, it provides a usefulestimate of the enthalpy that drives the formation of thisCu-rich amorphous phase. More precise measurementscould be obtained using Cu/Zr multilayers that amorphizecompletely and are devoid of excess Cu or Zr. Lastly, itis interesting to note that the formation enthalpy for thisamorphous phase is very close to that for the equilibriumcompound Cu51Zr14.

Figure 12. The heats of reaction 1 were calculated permole of atoms of amorphous Cu–Zr (34 at% Zr) thatformed in each sample. The values of 1H1 were obtainedexperimentally by subtracting the measured heats ofreaction 2 from the combined heats of reactions 1 and 2.The samples were isothermally annealed to completereaction 1 and then they were scanned in temperature toquantify the heat of reaction 2 [23].

3.3. The crystallization of amorphous alloys: Co–Siand Nb–Cu

Metastable amorphous alloys can also be formed in thinfilms through ion implantation. One example is found inthe field of silicides which play an important role in thefabrication of integrated circuits (ICs). Cobalt disilicide, forinstance, is being considered as a potential dopant diffusionsource for the formation of shallow junctions [41, 42].During IC fabrication, the silicide is amorphized bydopant implantation and then is crystallized by subsequentannealing in order to decrease its resistivity. Thiscrystallization process was studied recently using a seriesof co-deposited Co–Si alloys and DSC to determine howthe enthalpic driving force for crystallization varies withcomposition [43]. An enthalpy of−11.4± 0.5 kJ g-atom−1

was measured for the crystallization of the stoichiometricdisilicide and similar values were obtained with smallvariations in composition. This similarity suggests thatthe thermodynamic driving force for the transformation tothe crystalline state is relatively constant with composition.Therefore, small variations in composition during ICprocessing are unlikely to lead to variations in thecrystallization behaviour.

Other intriguing examples of metastable phases, onesthat are not obtainable in bulk form, are the amorphousalloys of Nb and Cu in the composition range 32–77 at%Cu [44]. Nb and Cu are immiscible in the equilibrium solidstate and have only limited miscibility in the liquid state.These immiscibilities are a reflection of the positive heatsof mixing in both states. Amorphous phases are frequentlyobserved in systems with largenegativeheats of mixing,but they are not expected in systems withpositiveheats of

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Figure 13. Formation enthalpies of amorphous Nb–Cufilms with various compositions, obtained by DSC. The fulllines are the enthalpies of formation of the BCC, FCC andamorphous phases, calculated by the CALPHAD methodfor the temperature of 425 ◦C [44].

mixing. Therefore, the formation of amorphous Nb–Cu isa unique example.

Plots of the power versus temperature for constant-heating-rate DSC experiments of amorphous Nb–Cuthin films showed relatively sharp exothermic peaksat approximately 400–450◦C. These exotherms wereassociated with the crystallization of the amorphous alloyinto the pure elements as detected by x-ray diffraction. Thecrystallization enthalpies were found to be in the range−4.5 to −7.6 kJ g-atom−1. By reversing the sign ofthese enthalpies, one obtains the positive heats of formationof these amorphous Nb–Cu alloys, referenced to purecrystalline Nb and Cu at the average reaction temperatureof 425◦C, in figure 13.

By extrapolating these experimentally measured en-thalpies to pure amorphous Nb and Cu and by assuming apositive heat of mixing in the amorphous phase, one obtainsunrealistically small or even negative formation enthalpiesfor pure amorphous Nb and Cu. Such values suggest thatthese amorphous pure elements would be the equilibriumphases instead of crystalline Nb and Cu. Clearly, this can-not be the case and therefore the assumption that the heat ofmixing of amorphous Nb–Cu is positive must be incorrect.

Figure 13 shows a parabolic fit to the experimentalenthalpy data for amorphous Nb–Cu alloys, which nowuses the low-temperature extrapolated enthalpies of liquidpure Nb and liquid pure Cu as enthalpies of the amorphouspure elements. (The use of extrapolated enthalpies ofliquids for amorphous phases, under the assumption thatthe amorphous phase is the low-temperature extrapolationof the liquid phase, has been successful in thermodynamicmodelling of a large variety of alloy systems to date[45].) As one can see in figure 13, the heat of mixingof the amorphous phase is now negative, as revealedby the concave curvature of the calculated amorphousenthalpy curve. However, as mentioned above, the formof the Nb–Cu phase diagram clearly implies that1Hmix ofthe liquid phase is positive at high temperatures. Thus,the heat of mixing of liquid Nb–Cu must be changingsign with undercooling [44]. This fascinating, new resultdemonstrates the unique contribution of DSC and thin

Figure 14. Formulation enthalpies of Cu–Cr alloy films withvarious compositions, obtained by DSC. The full lines arethe enthalpies of formation of the FCC and the BCC solidsolutions, calculated by the CALPHAD method for theaverage reaction temperature of 400 ◦C [46].

films to fundamental thermodynamic studies of alloys notobtainable in bulk form.

3.4. The formation of unstable solid solutions: Cr–Cu

The Cr–Cu system and the Nb–Cu system discussed aboveare similar in that they both exhibit negligible mutualsolubility in the solid phases and limited solubility in theliquid phase. However, the two systems differ in that, uponco-deposition, the Cr–Cu system forms supersaturated solidsolutions that are thermodynamicallyunstable[46] whereasthe Nb–Cu system formsmetastablemixtures of Nb and Cu[44]. The Cr–Cu alloys were sputtered onto liquid-nitrogen-cooled substrates in order to suppress decomposition duringdeposition [46]. X-ray diffraction studies indicated thatface-centred-cubic (FCC) solid solutions were formed forconcentrations up to 35 at% Cr and body-centred-cubic(BCC) solid solutions for Cr concentrations greater than 45at%. A gradual transition between these phases occurredin the range 35–45 at% Cr. The DSC traces of the Cr–Cu films revealed very broad exothermic reactions thatextended over almost the entire temperature range availablein a conventional DSC. The width of these traces wasapproximately 500◦C, in stark contrast to the relativelysharp exotherms that are usually found upon crystallizationof amorphous phases, as seen in figure 8 for Cu–Zr. Thisdifference in the DSC traces reveals the unstable natureof the Cr–Cu solid solutions, in contrast to the metastablenature of amorphous phases.

Figure 14 shows the heat released during thedecomposition of the Cr–Cu films as a function ofconcentration. Enthalpies of formation as large as+22.0 kJ g-atom−1 were obtained near the equi-atomiccomposition. These enthalpies can be applied tothermodynamic modelling by means of the CALPHADmethod and are exceedingly useful in predicting theformation of metastable and unstable phases. Typically,these predictions rely on extrapolations of thermodynamicfunctions from equilibrium to nonequilibrium compositionsand temperatures and, therefore, are often inaccurate. ThisCr–Cu example demonstrates how DSC data obtainedfrom thin film samples can fill gaps in the availability ofthermodynamic data.

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Figure 15. Experimental (broken line) and calculated (fullline) isothermal DSC traces for a Ge–31 at% Sn alloy at410 K [47].

3.5. Enthalpies of interfaces: Ge–Sn and Cu/Zr

Nanostructured materials typically have a high density ofinterfaces. The interfaces may be boundaries between finegrains in polycrystalline samples or boundaries betweenthin layers in multilayer films. These interfaces usuallyhave a higher enthalpy than do their surrounding phases andtherefore represent a source of excess enthalpy which canapproach several kJ g-atom−1 in very fine structures. Uponheating, nanostructures coarsen to reduce the overall freeenergy (and enthalpy) of the system. Although coarseningand grain growth are not ‘reactions’ in a chemicalsense, they do release enthalpy that can be measuredusing differential scanning calorimetry, as described inthe following two examples. In the first example weconsider the coarsening of a polycrystalline Ge–Sn alloywith nanometre-scale grains and in the second example weconsider the breakdown of layers in Cu/Zr multilayers.

3.5.1. The grain boundary enthalpy of Ge–Sn. Crystalline Ge–Sn alloys have shown promise assemiconducting infrared detectors [47, 48]. However, onlyGe–Sn solid solutions with compositions well beyond theequilibrium solubility limit of 1 at% Sn exhibit the desiredband structures. The lack of thermodynamic stabilityof these metastable solid solutions presents a processinglimitation when one attempts to increase the grain sizeto optimize the detection of infrared radiation. High-temperature anneals to increase the grain size can lead tophase separation and a degradation of the infrared detectioncapabilities. Here we present the measurement of thegrain boundary enthalpy that drives grain coarsening in ananocrystalline metastable alloy with 31 at% Sn that hadbeen fabricated by RF sputtering.

Figure 15 is an isothermal DSC trace taken at 410 Kfor this Ge–31 at% Sn film. Note that the rate of heat lossdecays with time as coarsening slows with increasing grainsize. The interfacial energy,Hint , stored in a sample with

an average grain sizer was modelled by Chen and Spaepen[49] and is given as

Hint = gγHV

r(6)

whereγH is the enthalpic part of the interfacial tension,Vis the molar volume of the sample andg is a geometricfactor equal to 1.3± 0.2 for equi-axed grains. The changein enthalpy with coarsening for this Ge–Sn alloy wasdetermined experimentally by integrating the heat flow withrespect to time in figure 15 and in a constant-heating-ratetrace (not shown) [47]. The integration yielded valuesof −1.26 ± 0.04 kJ g-atom−1 and −1.27 ± 0.04 kJ g-atom−1, respectively, for the change in interfacial enthalpy.By reversing the sign of these enthalpies, one obtainsthe magnitude of the enthalpy stored in the sample asinterfacial energy. The starting grain size was determinedto be 1.8 ± 0.2 nm from the width of x-ray diffractionpeaks. The molar volumeV was calculated assuming eightatoms per unit cell and a lattice constant of 0.5918 nm thatwas found from x-ray diffraction patterns of the crystallinealloy. Combining these values with equation (6) gaveγH = 119± 30 mJ m−2, assuming an infinitely large finalgrain size [47]. However, x-ray diffraction showed thatthe maximum grain size reached in the alloy prior to phaseseparation, when the alloy had been annealed at a constantheating rate, was only 4.5 ± 0.05 nm. Taking this finalgrain size into account, equation (6) now gives an interfaceenthalpyγH = 188±30 mJ m−2 for grain boundaries in thediamond cubic Ge–31 at% Sn alloy. This grain boundaryenthalpy is close to the value of 164 mJ m−2 reported for thegrain boundary energy of bulk crystalline Sn [50], raisingthe interesting possibility that the grain boundaries of theGe–Sn alloy are ‘Sn-like’.

3.5.2. The interface enthalpy in Cu/Zr multilayers.Nanoscale multilayer films can also contain significantquantities of interfacial enthalpy. In section 3.1,we considered DSC scans of Cu/Zr multilayers thattransformed into alternating layers of Cu/Cu51Zr14 througha series of two exothermic reactions. During the thirdreaction in figure 8, this nanoscale layered structurebreaks down and transforms to a non-layered, equi-axedstructure with fine Cu grains in larger grains of Cu9Zr2

(figures 9(c) and (d)). The tremendous growth of theCu–Zr intermetallic grains from approximately 20 nm inthe layered Cu/Cu51Zr14 structure (figure 9(c)) to 2000 nmin the equi-axed Cu and Cu9Zr2 structure (figure 9(d))releases considerable interfacial enthalpy and appears todominate the third exotherm in figure 8 [23]. Note that,unlike reactions 1 and 2, the total heat of reaction 3has no dependence on sample composition. This lack ofdependence is attributed to the presence of two sources ofheat in reaction 3: a transformation enthalpy,1Htr , and aloss of interfacial enthalpy,1Hint .

The interfacial enthalpy lost in reaction 3 can beapproximated using equation (6) and some simplifyingassumptions. Since the Cu phase has a relatively stablegrain size, its interfacial enthalpy is assumed to be constantduring the reaction. The final intermetallic phase, Cu9Zr2,

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Figure 16. The heats of reaction 3 from figure 8 are plottedversus the reciprocal of the Cu51Zr14 thickness. The heatswere calculated per mole of Cu9Zr2 that formed in eachsample.

has a relatively large grain size so its interfacial enthalpy isvery small and can be ignored. In contrast, the metastable,intermetallic phase, Cu51Zr14, has a very fine grain sizeand thus its interfacial enthalpy releases significant heatwhen lost and should be considered. This enthalpy canbe estimated by substituting the layer thickness,x, intoequation (6) in place of the average grain radius. Thus, theheat of reaction 3 can be written as

1H3 = 1Htr − g∗γVxCu51Zr14

(7)

where g∗ is a modified constant equal to 2.6 [51]. Theheat of reaction 3 was calculated per mole of Cu9Zr2

and the resulting heats are plotted versus the reciprocal ofthe Cu51Zr14 layer thickness in figure 16. Although thedata had some scatter,1H3 did increase linearly as thelayer thickness decreased, suggesting that more interfacialenthalpy was released in samples with thinner Cu51Zr14

layers [23]. Note in particular that the sample with thethinnest Cu51Zr14 layers (3.4 nm), released more than4 kJ/g-atom of interface enthalpy. This value is quitelarge, and it is similar in magnitude to the enthalpiesof transformations that are cited throughout this review.For example, the enthalpy produced in the reaction fromCu51Zr14 to Cu9Zr2 was determined to be only−0.93 kJ/g-atom based on equation (7) and the intercept in figure 16.

The slope of the line in figure 16 was used inconjunction with equation (7) to calculate an averageinterface enthalpy of 675± 40 mJ m−2. This value,which averages over Cu51Zr14/Cu51Zr14 grain boundariesand Cu/Cu51Zr14 interfaces, is close to the value of625 mJ m−2 [23, 50] reported for grain boundaries in pureCu. As nanostructured materials are developed for useat elevated temperatures, these large interfacial enthalpieswill play a significant role in determining the stability ofthese materials. With so few direct measures of interfaceenthalpies, DSC measurements such as these will proveexceedingly useful.

4. Kinetics

In addition to thermodynamic information obtained byintegrating a DSC trace, kinetic information can be obtainedfrom the shape of individual exotherms and from theheating-rate dependence of peak positions. Since thin filmfabrication typically allows one to obtain a large varietyof nonequilibrium phases, the kinetics of a broad rangeof reactions can be examined using thin film samples.For example, the kinetics of crystallization can be studiedusing amorphous films whereas the kinetics of grain growthcan be investigated with crystalline films that contain ahigh density of grain boundaries. In both cases, thereactions are driven by structural instabilities. One canalso investigate reactions that are driven by chemicalandstructural instabilities by depositing films with multiplelayers of different compositions. Indeed, multilayeredfilms have an ideal architecture for studying the formationof intermetallic phases from pure elements [52]. Bydecreasing the multilayer period, one can increase thedensity of interfaces at which the reaction begins. Thisenhances the thermal signal for the early stages of thereaction when the product phase nucleates and beginsto grow. In contrast, increasing the period enhancesthe thermal signal for the later stages of growth, whichin many cases is one-dimensional due to the layeredarchitecture of the film. In reviewing various DSC studiesof thin film reactions, we will begin with one-dimensionalgrowth which has a simple geometry and straightforwardmathematical analysis. Examples of the crystallization ofamorphous alloys, nucleation and growth of intermetallicphases and grain growth in nanocrystalline alloys willfollow.

4.1. One-dimensional growth

In this section we will discuss two types of one-dimensionalgrowth kinetics: 1D diffusion-controlled growth and 1Dinterface-controlled growth. Noteworthy examples ofcalorimetric studies of 1D growth reactions in thin filmsare the diffusion-controlled growth of amorphous phasesin Ni/Zr multilayers [53–55] and the interface-controlledepitaxial regrowth of amorphous Si and Ge produced byion irradiation [56, 57]. Since these results have alreadybeen reviewed in detail earlier [8] they will not be repeatedhere. Rather, we will refer to more recent investigationswhich also demonstrate the utility of calorimetry for thedetermination of 1D growth kinetics.

4.1.1. The one-dimensional diffusion-controlled growthof NiAl 3. Reactions in multilayer thin films often begin bythe formation of a contiguous product layer at the interfacebetween the reacting elements. The subsequent growth ofthis layer is typically normal to the interface and thereforeone-dimensional in nature. Here, we consider the case inwhich growth is limited by diffusion through the productlayer as it thickens. The rate at which heat is released bythis reaction in a DSC experiment, dH/dt , is assumed tobe proportional to the rate at which the product phase is

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Figure 17. The DSC trace of a 1Ni/3Al multilayer film with3 = 2 µm period, taken at a heating rate of 40 K min−1.Using equation (8), the rate of heat flow was converted intoa growth rate, dx/dT . The full line is the best fit of 1Ddiffusion-controlled growth kinetics to the data points in thetemperature range 600–760 K. The inset shows a Kissingerplot of the peak-temperature shift as a function of theheating rate [58].

thickening, dx/dt . For a constant-heating-rate experimentthe thickening rate is given by

dx

dt= β dx

dT= xmax

1H

dH

dt(8)

where x is the product phase thickness,β is the heatingrate,1H is the enthalpy change of the reaction andxmaxis the maximum product phase thickness which is typically3/2. The exothermic growth of NiAl3 in Ni/Al multilayershas been studied using DSC and equation (8) [58]. Figure17 shows a DSC trace obtained from a 1Ni/3Al multilayerwith 3 = 2 µm. In this trace, the small peak at about550 K is due to the initial nucleation and growth of NiAl3,which will be discussed in section 4.2. The major portionof the trace, however, is characterized by the 1D diffusionalgrowth of this product phase. The reaction ends abruptlywhen the reactant phases are consumed simultaneously atall locations at the interface.

The 1D diffusional growth rate can also be written asx = dx/dt = kd/x, wherekd is the rate constant whichis proportional to the interdiffusion coefficient. Assumingthat kd can be described by an Arrhenius ansatz, it followsthat

xx = kd = kd,0 exp

(− Ed

kBT

)(9)

where kd,0 and Ed are the pre-exponential factor andthe activation energy for diffusion-controlled growth,respectively, andkB is Boltzmann’s constant. Underisothermal conditions,kd is constant and integration ofequation (9), rewritten asx dx = kd dt , leads to thewell-known parabolic growth lawx2/2 = kdt for whichthe thickening rate decreases with time. For constantheating rates, integration of equation (9) yields [58, 59] (seeappendix A)

x2

2= kd,0

β

kBT2

Edexp

(− Ed

kBT

)(10)

and the thickening rate continuously increases withtemperature. The derivative of equation (10), dx/dT ,was fitted to the experimental data in figure 17 and theactivation energyEd and the pre-exponential factorkd,0were calculated from this fit to be 1.72 eV and 2×10−3 m2 s−1, respectively. It is important to note that this1D diffusion-controlled model predicts the proper shape ofthe DSC trace in figure 17 for the Ni/Al system, but, as wewill see later, it cannot do so for the Ti/Al system in whichgrowth appears to be interface controlled.

The activation energy for 1D diffusion-controlledgrowth can also be obtained using the Kissinger analysisthat relates the peak temperatures to the heating rate[60, 61]. The validity of the Kissinger method can be seenby rearranging equation (10) to give

β

T 2= 2kBkd,0

x2Edexp

(− Ed

kBT

). (11)

The peak temperature,Tpeak, at which the reactant phasesare consumed will shift according to equation (11) whendifferent heating rates are used and a Kissinger plot ofln(β/T 2

peak) versus 1/(kBTpeak) will give a straight line withslope−Ed . The quantitykd,0 can be obtained from theintersection with the abscissa. The inset in figure 17 showsthe dependence ofTpeak on the heating rate for the 1Ni/3Almultilayer films, presented in a Kissinger plot. From thisplot, Ed = 1.64 eV andkd,0 = 6.5× 10−4 m2 s−1 wereobtained, in good agreement with the values deduced above.

To demonstrate the consistency between a 1D diffusion-controlled growth model and the Ni/Al DSC data further,an Arrhenius analysis was performed. As can be seen fromequation (9), a plot of ln(xx) = ln(kd) versus 1/(kBT )should yield a straight line with slope equal to−Ed .x(t) is obtained directly from the original DSC trace andequation (8) andx(t) is obtained by cumulative integrationof x(t). Figure 18 demonstrates that this Arrhenius plot is infact linear over the majority of the reaction’s temperaturerange. In addition, all DSC traces obtained at differentheating rates fall onto a single straight line, confirmingthe consistency between the model and experiment. ThevaluesEd = 1.64 eV andkd,0 = 5× 10−4 m2 s−1 wereobtained from this plot and are in good agreement with theabove results. Note that data from any one of the DSCtraces in figure 18 can be used to determine the growthconstant,kd , which varies by four orders of magnitude ina temperature interval of approximately 200 K. In contrast,in conventional diffusion couple studies the determinationboth of the activation energy and of the pre-exponentialfactor requires a large number of experiments with a varietyof annealing times and temperatures. Finally, one shouldalso note that this Arrhenius method of analysis and theprevious two methods yield strikingly similar results for1D diffusional growth of NiAl3.

4.1.2. One-dimensional interface-controlled growth ofTiAl 3. The 1D growth of TiAl3 has also been studiedusing DSC and multilayer samples [62]. A typical DSCtrace for a 1Ti/3Al multilayer film is shown in figure 19with the exothermic heat flow converted into a growth

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Figure 18. DSC traces of 1Ni/3Al multilayer films with3 = 2 µm performed at various heating rates, presented asan Arrhenius plot of xx = kd versus 1/(kB T ) [58].

rate using equation (8). The DSC trace shows a smallpeak at 750 K due to a transformation from a metastableL12 structure into a second metastable D023 structure forTiAl 3 [63]. However, most of the trace is dominatedby an asymmetrical peak ending at approximately 900 K.The smooth rise and the abrupt drop for this exothermare indicative of a one-dimensional growth process thatends suddenly as reactants are consumed. As illustrated infigure 19, analysis of the DSC trace shows that a diffusion-controlled growth model does not yield the proper shape ofthe exotherm. Driven by this disagreement, isothermal DSCand x-ray diffraction experiments were also performed onthese samples, which revealed that the growth rates wereapproximately constant over a significant period of time.This is in contrast to diffusion-controlled growth, for whichthe growth rate would decay proportionally to 1/

√t , and

suggests that growth of TiAl3 is interface controlled. Thefollowing equation gives the temperature dependence of the1D interface-controlled growth rate:

x = ki = ki,0 exp

(− Ei

kBT

)(12)

where ki is the rate constant andki,0 and Ei are thecorresponding pre-exponential factor and the activationenergy, respectively. A fit of equation (12) withEi =1.89±0.05 eV yielded a better agreement with the constant-heating-rate data in figure 19 than did equation (8) fordiffusion-controlled growth. The activation energy of1.89 eV was determined by the Kissinger method. (Thevalidity of the Kissinger method for 1D interface-controlledgrowth can be shown by analogy with section 4.1.1.)

The reasons why the growth of TiAl3 is betterdescribed by interface- than by diffusion-controlled growthkinetics are not yet clear. Parallel electron microscopyinvestigations, however, have shown that TiAl3 is formedby repeated nucleation and growth at the Ti interface, ratherthan by pure growth [64]. It was therefore suggestedthat the grain structure evolution during TiAl3 formationaffected the growth kinetics [65]. However, regardlessof the source of interface-controlled growth, the twoabove examples demonstrate the outstanding capabilities ofmultilayer film DSC studies in testing the validity and indetermining the details of an anticipated reaction process.

Figure 19. The DSC trace of a 1Ti/3Al multilayer film with2 µm period taken at a heating rate of 40 K min−1,converted into growth rate units. The full and the brokenlines were calculated using interface- anddiffusion-controlled growth kinetics, respectively, and theexperimentally determined activation energy of 1.89 eV forboth processes. The corresponding pre-exponential factorswere calculated on the basis of the experimental result thatthe reactant materials were consumed at the peaktemperature of 907 K, namely x(907 K) = 1 µm. Thisyielded ki ,0 = 6× 1011 nm s−1 and kd ,0 = 3× 1014 nm2 s−1

[62].

4.2. Nucleation and growth

A thin film transformation is expected to occur by nucle-ation and growth when the material under investigationis in a metastable state due to the presence of nucleationbarriers. In this section we describe three different typesof nucleation-and-growth transformations. The first is thecrystallization of an amorphous Co–Si alloy and the secondand third are the early stages of tri-aluminide formation inNb/Al and Ti/Al multilayer films.

4.2.1. The crystallization of amorphous Co–Si alloys.As mentioned in section 3, amorphous Co–Si alloys areof interest due to their potential use as dopant diffusionsources for forming shallow junctions in integrated circuits.Details of the crystallization process of amorphous Co–Sialloys such as the crystallization temperature and grainsize distribution are expected to play a critical role indetermining the uniformity of dopant out-diffusion [66].Thus understanding the kinetics of the crystallization isimportant in optimizing the out-diffusion process. For thecrystallization example cited here we demonstrate that DSCstudies yield the activation energy for growth, the Avramiexponent – which embodies information on the nucleationand growth mechanisms and growth morphologies – andthe density of nuclei.

Amorphous Co0.29Si0.71 films crystallize in the temper-ature range 170–220◦C in constant-heating-rate DSC scans,as shown in figure 20(a) [43]. The shifts in peak temper-ature with heating rate were used in a Kissinger analysisto yield an activation energy of 1.16± 0.07 eV. This valueagrees with three other measurements of activation ener-gies for the crystallization of CoSi2. 0.93± 0.10 eV wasmeasured using electrical conductance [67]; 1.17±0.03 eVwas measured using time-resolved reflectivity and Ruther-ford back scattering that was combined with channelling

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(a)

(b)

Figure 20. (a) Constant-heating-rate DSC traces ofCo0.29Si0.71 samples heated at 2.5, 5, 10 and 20 ◦C min−1.(b) Isothermal DSC traces of Co0.29Si0.71 samples annealedat 148 and 153 ◦C. Reprinted with permission from [43].Copyright 1992 American Institute of Physics.

[68]; and lastly, 1.1 eV was obtained by measuring theradii of crystalline grains of CoSi2 during in situ annealingin a high-voltage electron microscope [69].

In order to quantify the crystallization process further,isothermal DSC scans of amorphous Co0.29Si0.71 wereperformed and the resulting data were analysed using theJohnson–Mehl–Avrami (JMA) theory [70–74]. As shownin figure 20(b), these DSC measurements produced bell-shaped peaks, indicative of a nucleation-and-growth-typeprocess [49, 75]. These can clearly be distinguished from1D growth kinetics, in which growth rates either remainconstant or decay with time (see appendix B), and alsofrom grain-growth kinetics (as shown below), which yieldDSC traces that decay with time. The transformed volumefraction,X, at constant temperature is given by the JMAtheory for nucleation and growth [74],

X(t) = 1− exp(−ktn) (13)

where k = k(t) is a temperature-dependent rate constantand n is the Avrami exponent. The Avrami exponentn was obtained for the crystallization of Co0.29Si0.71 byassuming that the heat flow rate is proportional to the

transformation rate, by normalizing and integrating thepeak and by plotting the data as an Avrami plot, namelyas ln[− ln(1 − X)] versus ln(t). Such a plot shouldgive a straight line with a slope equal ton according toequation (13). For the Co–Si datan was found to be3.1 ± 0.1. For the analysis it was necessary to includean incubation timeτ into the Avrami equation (13), byreplacingt by t − τ . The value ofτ was adjusted until thebest fit to the data had been obtained.

According to the JMA theory there are essentiallytwo processes that could lead to an exponent ofn = 3.One is theformation of nuclei at a constant ratefollowedby the growth of these nuclei at a constant velocity intwo dimensionsand the other is the presence of afixednumber of nucleiwhich grow at a constant velocity inthree dimensions. To differentiate between these twocases, the crystallization of an amorphous Co0.33Si0.67

sample was observedin situ in a transmission electronmicroscope. Figures 21(a) and (b) are the micrographsof partially and fully crystallized states, respectively.The number of crystalline grains remained approximatelyconstant throughout the transformation, indicating eitherthat the crystal nuclei were ‘pre-existing’ ones or thatthe number of crystal nuclei saturated early into thetransformation. Furthermore, the grain size did notreach the film thickness until close to the end of thecrystallization process, indicating that the crystals weregrowing three-dimensionally throughout the transformation.Grain diameters increased linearly as a function of time,with a growth velocity ofv = 4.4±0.5 nm min−1 at 161◦C.All of the above information indicates that the interpretationof n is one involving a fixed number of nuclei and interface-controlled growth of grains in three dimensions.

The number of nucleiN per unit volume wasdetermined through a measurement of the kinetic constantk and a calculation of the growth velocity at a giventemperature [66]. The kinetic constant for the abovenucleation conditions, growth morphology and mechanismis given by

k = 43πv

3N. (14)

The calculated density of nuclei remained almostunchanged at 1020 m−3 in the range 140–170◦C and agreedwell with the density determined fromin situ TEM studiesat 161◦C. The combination of the activation energy, theAvrami exponent and the density of nuclei gave a relativelycomplete description of the crystallization of these Co–Si films. This analysis also demonstrates the utility ofaugmenting DSC studies with TEM observations.

4.2.2. The early stages of reaction in Nb/Aland Ti/Al multilayer thin films. One of the mostinteresting features of reactions in multilayer films is thephenomenon of phase selection in which product phasesappear sequentially rather than simultaneously and certainequilibrium phases are absent whereas metastable phasesare commonly observed. (See appendix B for additionaldiscussion.) A prominent example is the formation ofamorphous phases in multilayer diffusion couples, whichwas discussed in section 3.2. Since the prediction of

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(a) (b)

Figure 21. In situ TEM bright-field images of a Co0.33Si0.67 sample 100 nm thick annealed at 161 ◦C: (a) partially crystallizedand (b) fully crystallized.

phase formation at interfaces is extremely important inmany technological applications, it has prompted numerousinvestigations. However, there is still no universallyaccepted theory and predicting phase selection remains ascientific challenge.

An intriguing observation that was first found bycalorimetry and brought a new viewpoint into this fieldwas the fact that the formation of the first product phasein a multilayer film can be a two-stage process [65, 76–84]. As an example, figure 22 shows heat flux DSCtraces for the formation of NbAl3 in sputter-depositedNb/Al multilayer films with various Al layer thicknesses[25, 76, 79]. Extensive x-ray and electron diffraction studiesverified that both peaks A and B were associated withthe formation of the single product phase, NbAl3. Notethat the relative areas of the peaks changed with Al layerthickness in a systematic way. Peak A dominated for smallthicknesses and peak B became increasingly pronouncedfor large thicknesses. Similar two-peaked traces have alsobeen observed for evaporated Nb/Al multilayer thin films[25, 78, 82], as well as for other systems such as Ni/Al[26, 27, 80], Ti/Al [63, 64], Ni/amorphous Si [81, 82] andV/amorphous Si [83, 84]. In all cases the two peakscorrespond to the formation of a single phase.

The formation of a single phase in two stages isinconsistent with any combination of 1D growth processes,as discussed in appendix B, but it is explained by the modelof Coffey et al for a two-stage transformation [82]. In thismodel, a product phase first nucleates and grows laterally inthe plane of the interface and forms a coalesced, contiguouslayer in stage one (peak A). In stage two this contiguouslayer grows normal to the interface until one or bothreactants are consumed (peak B). Both stages are shownschematically in figure 23. The transformation associatedwith peak A (stage one) is described by a JMA nucleationand growth process which, under isothermal conditions, isgiven by equation (13). Application of the JMA equationto constant-heating-rate conditions [26, 75] leads to thefollowing equation for the volume fraction transformed:

X(T ) = 1− exp

[−(k0kBT

2

βE

)nexp

(− nE

kBT

)](15)

Figure 22. Heat-flux DSC traces for a series of Nb/Almultilayer films annealed at 20 K min−1. The periodicity, 3,the Al layer thickness, xAl , and the Nb/Al ratio are listed oneach trace. The traces were limited to a maximumtemperature of 900 K and thus do not include the reactionsthat occur beyond this temperature for films with Nb:Alratios greater than 1:3. Note that the height of each tracehas been adjusted to make the shape of the trace moreclear. The broken lines were calculated using the model ofCoffey et al [82]. Details of the calculation were given in[76].

where k0 and E are the pre-exponential factor and theactivation energy of the JMA process. dX/dT is thenproportional to the heat released in peak A in the DSCtraces. Coffeyet al [82] assumed that the transformationbegan with a fixed number of pre-existing nuclei inthe plane of the interface which subsequently grew as

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Figure 23. A schematic drawing of stages one and two forthin film reactions, using the example of NbAl3 formation inNb/Al multilayer thin films.

cylindrical grains. The growth parallel to the interface wastaken to be interface controlled and was represented by theJMA equation withn = 2. The subsequent thickeningof the layer, peak B in the DSC traces, was modelled by1D diffusion-controlled growth, equation (10). Calorimetryexperiments (isothermal and constant-heating-rate ones) canbe used to determine all the model parameters separately,except for the nucleation site density per unit area,N , andthe radial growth prefactor,ki,0, which appear asNk2

i,0[82]. Determination of these two parameters requires directobservations of the evolution of the product phase grainsby a technique such as transmission electron microscopy.

The broken lines in figure 22 represent the traces forthe formation of NbAl3 simulated on the basis of the abovemodel. As can be seen in figure 22, the simple model ofCoffey et al successfully describes the outstanding features,namely, the two peaks and their systematic dependenceon the layer thickness. This model has been successfulin describing the behaviour of multilayers of Nb/Al (firststudied by Coffeyet al [82]), and also in Ni/Al [80],

Figure 24. The DSC trace of a 1Ti/3Al multilayer film with3 = 10 nm, taken at a heating rate of 40 K min−1. The fullline is the best fit of a JMA process to the measurement[63, 75].

Ni/amorphous Si [82] and V/amorphous Si [83].However, although the simple model of Coffeyet al

[82] describes the essential features of the DSC traces,the model underestimates the widths of peaks A and B insome cases. Underestimating the width of peak A indicatesthat n is less than 2, which has been found for severalsystems using isothermal and constant-heating-rate DSCexperiments [27, 63, 75–79]. Underestimating the widthof peak B may be attributed to nonuniformities in layerthickness which have been observed in many multilayersystems [79]. These variations in thickness prevent anabrupt termination of the reaction and an abrupt end topeak B [65, 85].

As an example of a lower Avrami exponent producingbetter fits to peak A of the experimental data, a 40 K min−1

DSC trace of a 1Ti/3Al multilayer with 10 nm period isshown in figure 24 [63, 75]. Equation (15) was fitted to thisdata using an activation energy of 1.54± 0.05 eV obtainedby Kissinger analysis and an Avrami exponentn = 1.3.As can be seen in figure 24, the fit to the DSC data isextremely good. Additional isothermal DSC measurementsconfirmed that the value ofn was close to unity [63, 75],thus demonstrating the utility of equation (15) in modellingpeak shapes of JMA processes. These lower values ofn may be a consequence of the spatial bias, namely thedeviation from spatial randomness, in the nucleation of theproduct phase [76, 86]. Confirmation of this explanationrequires direct observations of grain structure evolution andhas not yet been obtained.

Although the model of Coffeyet al could benefit frommodifications, it has been a significant step forwards inunderstanding the details of compound formation in thinfilms by establishing that nucleation, as well as growth,must be included in the description of these reactions.DSC studies have played a pivotal role in prompting andverifying this understanding. These studies also suggestthat nucleation is likely to play a part in the selection ofphases in thin film reactions. In other words, there must bebarriers to nucleation of the first phase. This is unexpected,because the system contains very large driving forces forfirst phase formation [87]. This inconsistency has prompted

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Thermodynamics and kinetics of thin film reactions

new investigations of the true driving forces for first-phaseformation [88, 89].

4.3. The grain growth in Ge–Sn alloys

In section 3.5.1 we demonstrated how DSC traces could beused to determine the enthalpic part of the grain boundaryenergy in Ge–Sn thin film alloys. Here, we show how thesame traces can be used to determine the kinetics of graingrowth in this material, namely the values ofkg(T ) andpin the following equation for the average grain radius,r(t):

rp(t) = rp0 + tkg(T ) (16)

wherer0 is the initial grain radius. Using this equation andequation (6), the value of the grain growth exponent,p,can be found from the following equation:

HT (t) = (H0)T

(t + ττ

)−1/p

(17)

whereτ is a time constant defined as(r0)p

T /kg(T ). τ acts asa fitting parameter for obtaining a straight line with slope−p, when ln(t + τ) is plotted versus lnHT (t). For theisothermal DSC data at 410 K (figure 15) and 450 K (notshown), the values ofτ were 3.4–3.5 and 4.2–4.8 minand the slopes of the lines,−p, were−1.76 and−1.6,respectively. The average value ofp was determined tobe 1.7± 0.1, which compares well with the value of 2 instandard grain growth models [90–93]. The full line infigure 15 is a simulated trace using the values ofp = 1.76andτ = 3.4 min.

The values ofp can also be used to determine theactivation energy for grain growth,Eg, by assuming anArrhenius temperature dependence for the rate constant,kg(T ) = kg,0 exp[−Eg/(kBT )] and by letting t1/2 be thetime at which the total enthalpy of the system has decreasedto half its initial value. Then the following product,t1/2H

p

0 kg,0 exp[−Eg/(kBT )], is constant. Using the datafor 410 and 450 K,Eg was found to be approximately0.5 eV for the Ge–Sn system. This activation energy forgrain growth is low compared with the value of 2.2 eVfound by Chen and Spaepen [49] for grain growth inmicroquasicrystalline Al–Mn thin films. Clearly, moreisothermal traces should be used to determinep and Egaccurately [94]. As a final point, note that the decayingisothermal trace in figure 15 confirms that the Ge–Snalloy was nanocrystalline at the start of the reaction anddid not nucleate from an amorphous phase [49]. Thiscontrasts sharply with the ‘bell-shaped’ traces for theamorphous cobalt silicide films in section 4.2.1, whichindicate that a transformation from one phase to another istaking place which involves both nucleation and growth.This qualitative conclusion that the Ge–Sn alloy wasnanocrystalline and the quantitative measures of graingrowth parameters for this Ge–Sn alloy, illustrate how onecan use DSC to study grain growth kinetics in thin films.

5. Summary

We have reviewed a broad range of differential scan-ning calorimetry experiments that investigated the thermo-dynamics and kinetics of thin film reactions. Whereassome of these reactions are directly relevant to commercialapplications, others are designed to address basic scientificquestions, a number of which cannot be addressed usingconventional bulk materials. We began the review withdescriptions of differential scanning calorimetry and sam-ple preparation techniques in order to provide the readerwith some background on experimental methods. We thendemonstrated how DSC can be used to measure heatsof formation, enthalpies of crystallization and enthalpiesof interfaces in homogeneous and in layered thin films,emphasizing when possible the unique benefits of combin-ing DSC and thin film samples. In the final sections of thepaper, we reviewed how kinetic parameters of nucleation,growth and coarsening could also be determined from DSCtraces. We hope that these examples and their discussionwill foster increased utilization of DSC and thin film sam-ples for investigating the thermodynamics and kinetics oftransformations that are critical to the application of mate-rials both in thin film and in bulk form.

Acknowledgments

KB gratefully acknowledges support under NSF DMR-9308651, NSF DMR-9458000 and Deutsche Forschungsge-meinschaft SFB371 and the donation of a 1600◦C furnaceby TA Instruments. TPW gratefully acknowledges supportunder AFOSR F496209610147, ARL DAAL 019620047,NSF DMR 9702546 and the NSF’s MRSEC on Nanostruc-tured Materials at Johns Hopkins University.

Appendix A. Integration of kinetic rate equationsunder constant-heating-rate conditions

Using the 1D diffusional growth as an example, we showin this section how a variety of kinetic rate equations canbe integrated under nonisothermal conditions to yield thegrowth rates for constant-heating-rate experiments. Thisallows one to compute the corresponding DSC traces andto fit the resulting equations to experimental data.

The rate equation for 1D diffusional growth, equa-tion (9), can be rearranged as

x dx = kd dt = kd,0 exp

(− Ed

kBT

)dT

β. (A1)

The integral of the right-hand side of this equation isthe so-called temperature integral which cannot be solvedanalytically. However, a variety of good approximationsexist, as discussed in detail elsewhere [95]. In this paperwe use the approximation given by Coats and Redfern [96],∫

exp

(− E

kBT

)dT ≈ kBT

2

Eexp

(− E

kBT

). (A2)

The error in this approximation is usually well below10%. Approximations with a higher level of accuracy were

3183

C Michaelsen et al

reviewed in [95]. With equation (A2), equation (A1) canbe integrated to yield equation (10), that is

x2

2= kd,0

β

kBT2

Edexp

(− Ed

kBT

)(A3)

so that

x =[

2kd,0kBβEd

T 2 exp

(− Ed

kBT

)]1/2

. (A4)

Inserting equation (A4) into equation (A1) leads to

dx

dT=(kd,0Ed

2kBβ

)1/2 1

Texp

(− Ed

2kBT

). (A5)

As demonstrated in figure 17, equation (A5) can be usedfor direct fits to experimental DSC traces in order to obtainkd,0 andEd from a single measurement. Alternatively, aplot of T dx/dT versus 1/(kBT ) should give a straight linewith slope−Ed/2.

Appendix B. The model of G osele and Tu:linear-parabolic 1D growth

Gosele and Tu [97–99] developed a model that combinesinterface-controlled and diffusion-controlled 1D growth inorder to explain the phenomena of phase selection andmetastable phase formation that are frequently observedduring the early stages in multilayer reactions. Theirmodel is often used to explain phase selection in studiesthat utilize techniques other than DSC such as TEMand Rutherford back scattering [100]. We discuss thismodel now to demonstrate clearly that inputting physicallyrealistic numbers into their model cannot lead to twoseparate DSC peaks in constant-heating-rate traces for theformation of a single phase, as seen in figure 22.

An appropriate rate equation for their combined 1Dgrowth process is [18, 97–99]

x =(k−1i +

(kd

x

)−1)−1

= ki

1+ kix/kd (B1)

which corresponds to two growth processes occurring inseries. Forkix � kd equation (B1) reduces to the entirelyinterface-controlled process and forkix � kd it reducesto the entirely diffusion-controlled process. This showsthat the kinetics of such a combined process are controlledby the slower one of the two and that the faster onecan be ignored when there are large differences in therate constants. Equation (B1) can be integrated to obtainthe dependence of the thickness on time, which upon re-insertion into equation (B1) yields

x = kd(2kdt + k2

d

k2i

)1/2 . (B2)

Equation (B2) is plotted as the inset in figure 25 for variousvalues ofkd/ki . As can be seen in figure 25, the growthrate follows a 1/

√t dependence forkd/ki = 102 nm, as

expected for an entirely diffusion-controlled process, and

Figure 25. Growth rates for a linear–parabolic process at aconstant heating rate of 40 K min−1, calculated bynumerical integration of equation (B1) withEi = Ed = 1.9 eV, ki ,0 = 6× 1011 nm s−1 andkd ,0 = 3× 1014 nm2 s−1 (kd/ki = 500 nm). These kineticparameters were chosen on the basis of the DSC trace forthe Ti/Al multilayer film shown in figure 19. The solelyinterface-controlled and the solely diffusion-controlledprocesses were also calculated for comparison. The insetshows the isothermal growth rates for a linear–parabolicprocess, calculated using equation (B2). In this plot thekinetic parameters kd and ki were adjusted for each curveto yield a product phase 1 µm thick at the end of a 1000 sanneal.

becomes constant forkd/ki = 104 nm, as expected for anentirely interface-controlled process. It can be concludedfrom figure 25 that both kinetic processes contribute to theDSC trace only whenkd/ki lies within a relatively narrowrange. The value ofkd/ki at which the change-over fromdiffusion-controlled to interface-controlled kinetics occursdepends on the thickness range considered.

When equation (B1) is solved numerically underconstant-heating-rate conditions, both processes becomeapparent in the DSC trace only when their activationenergies are approximately equal, namely, whenEi ≈ Ed .A difference in the activation energies of typically only0.2 eV is sufficient to let the process with the higheractivation energy dominate. For the case in whichEi ≈Ed , a DSC trace initially follows the interface-controlledprocess and then continuously approaches the diffusion-controlled process during the growth of the product phase,as shown in the main part of figure 25. Note that there isno evidence of a double peak and that varying the values ofactivation energies (or rate constants) cannot produce twoexotherms for the formation of a single phase.

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