Competition between tensile and compressive stress mechanisms duringVolmer-Weber growth of aluminum nitride filmsBrian W. Sheldon, Ashok Rajamani, Abhinav Bhandari, Eric Chason, S. K. Hong et al. Citation: J. Appl. Phys. 98, 043509 (2005); doi: 10.1063/1.1994944 View online: http://dx.doi.org/10.1063/1.1994944 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v98/i4 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Competition between tensile and compressive stress mechanismsduring Volmer-Weber growth of aluminum nitride films

Brian W. Sheldon,a� Ashok Rajamani, Abhinav Bhandari, Eric Chason, S. K. Hong, andR. BeresfordBrown University, Division of Engineering, Providence, Rhode Island 02912

�Received 6 July 2004; accepted 13 June 2005; published online 18 August 2005�

Stress evolution during molecular-beam epitaxy of AIN films was monitored with in situ curvaturemeasurements. Changes in the growth rate produced large stress variations, with more tensile stressobserved at higher growth rates. For example, at a growth temperature of 750 °C the instantaneoussteady-state stress in films with similar grain sizes varied from −0.15 GPa at a growth rate of90 nm/h, to approximately 1.0 GPa at a growth rate of 300 nm/h. To explain these results, wedevelop a kinetic model of stress evolution that describes both tensile and compressive mechanisms.The tensile component is based on a mechanism which is proposed here as an inherent feature ofgrain-boundary formation. The compressive component is based on our recent model of atominsertion, driven by the excess chemical potential of surface adatoms that is created by the growthflux. The combined model predicts that the stress is largely governed by the competition betweentensile and compressive mechanisms, which can be conveniently described with a single parameter,�. The limiting values �→0 and �→ +� correspond to previous models of compressive and tensilestresses, respectively. © 2005 American Institute of Physics. �DOI: 10.1063/1.1994944�

I. INTRODUCTION

Residual stresses in thin films and coatings can have asignificant impact on the reliability of the electronic devicesand structural components in which these materials are used.Film stress is comprised of a thermal component �due todifferences in thermal expansion� and an intrinsic componentdue to the growth process. Of the two, intrinsic stresses areless well understood and can be caused by a variety ofmechanisms. In the Volmer-Weber film growth, individualislands that first form on a substrate usually exhibit compres-sive stress. A shift to tensile stress is often observed whenthese islands grow together to produce a continuous film; thisis often followed by a return to compressive stress at longergrowth times.1–3 Recent literature contains a variety of ideasabout the mechanisms that produce these stresses.2–16

The tensile stress caused by island coalescence was char-acterized by Hoffman and co-worker, who proposed that theelastic strain energy introduced by this process was driven bythe energy reduction associated with creating a grain-boundary segment from the solid-vapor surfaces of twoneighboring grains.17,18 This early work proposed that thesesurfaces will pull together to form a grain boundary when thegrains are separated by a sufficiently small critical distance.Several recent analyses of this phenomenon describe tensilestress as a function of ��=�I−2�, where �I is the free en-ergy per area of the formed grain boundary and � is the freeenergy per area of the solid/vapor surfaces prior to coales-cence. Nijhawan et al. used a simple energy balance to pre-dict a maximum stress: �max= �−2M�� /L�1/2, where M is thebiaxial modulus of the film and L is the grain size.5 With thetype of geometry in Fig. 1�a� Nix and Clemens obtained a

similar expression based on a more detailed analysis of the“zipping” process for the formation of an interface betweentwo surfaces joined at a cusp.6 Freund and Chason extendedthis approach to consider initial contact mechanics for a va-riety of configurations.7 Both the Nix-Clemens and Freund-Chason approaches use what is essentially a Griffith criteria,expressed here as

��US + UE��b

= �� + Gb = 0, �1�

where US and UE are the excess energies associated withsurfaces/interfaces and elastic strain, respectively. Equation

a�Electronic mail:brian�sheldon@brown.eduFIG. 1. �a� Model film configuration at the point where islands first impinge.�b� Model film configuration after additional growth has occurred.

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�1� describes the variation of these energies with respect to asmall incremental change in the length of the grain boundary,�b. By analogy with fracture mechanics, Gb is the energyrelease rate �or the crack extension force at the top of thegrain boundary�. The value of Gb depends on the stress dis-tribution in the entire film-substrate system. This quantityhas dimensions of energy per area, and has been widely stud-ied in fracture mechanics, with analytical descriptions avail-able for certain configurations.19,20 Finite element models�FEMs� have also been used to apply Eq. �1� to both theinitial contact stress8,9 and continuing tensile stress evolutionafter the islands initially impinge.10,11

Compressive stresses observed during postcoalescencegrowth are also the subject of recent debate. One proposedmechanism is that inserting extra atoms at the grain bound-aries can lead to compressive stress.6 The thermodynamicdriving force for this type of process has been expressed interms of the adatom supersaturation that is created by thegrowth flux.12,13 Two other possibilities that do not requiregrain boundaries are Spaepen’s suggestion that adatoms canbe incorporated at ledge sites on the growth surface,14 andThompson and co-workers who propose that an adatom su-persaturation on the growth surface can significantly increasethe compressive surface stress.15,16

Most of the models cited above predict limiting valuesfor the stress in the film, based on energy considerations. Forexample, the energy trade-off in Eq. �1� has been used toobtain expressions for the tensile stress that obey the generalform7,9,10

��� = AE�− ��

EL�B

, �2�

where A and B are constants that depend on the dimension-ality of the islands, and E is Young’s modulus. For compres-sive stress generated by the insertion of excess atoms at thegrain boundaries, the thermodynamic limit on the stress isassociated with the adatom supersaturation on the growthsurface, ss,

13

��� =− kBT

�ln sS, �3�

where � is the atomic volume of the solid and kB is theBoltzmann constant �the kinetics of this process arediscussed further in Sec. III�. Analogous thermodynamiclimits can also be obtained for other compressive stressmechanisms.

It is often assumed that the tensile mechanism operatesonly during initial island coalescence; however, continuingtensile stress evolution after initial coalescence has been ob-served in a diamond formed by chemical-vapor deposition�CVD�,21 in electrodeposited films,22 and in the AIN filmsdescribed in Sec. II. With this in mind, the model developedin Sec. III is designed to reconcile the observed evolution ofboth tensile and compressive stresses after island coales-cence. The existing models for both mechanisms are essen-tially energetic analyses relating the stress to thermodynamicdriving forces, as seen in Eqs. �2� and �3�. However, the

kinetic competition between tensile and compressive stressevolutions must be considered to explain our experimentalobservations.

II. ALUMINUM NITRIDE FILMS

In films where postcoalescence tensile and compressivemechanisms are both operating, the stress should fall be-tween the respective energetic limits �e.g., Eqs. �2� and �3��.To examine this situation we consider AlN films that weregrown on Si �111� substrates at 750 °C, by an electron cy-clotron resonance molecular-beam epitaxy �MBE� process.In these experiments the film growth rate was varied bychanging only the Al flux. Further details of the growth pro-cess and film characteristics are described elsewhere.23 Thedeposits are epitaxial with respect to the substrate, as evi-denced by reflected high-energy electron-diffraction�RHEED� patterns obtained in the growth chamber. How-ever, the large lattice mismatch �19%� and relatively lowatomic mobilities lead to Volmer-Weber growth where themisfit strain relaxes prior to island coalescence, with misori-entation boundaries between neighboring grains and grainsizes of 40–80 nm.24 The grain size and roughness at the topsurface of the films were examined with scanning electronmicroscopy �SEM� and atomic force microscopy �AFM�.The SEM image in Fig. 2 shows the top surface of a film,where the surface topography reflects the underlying grainstructure. In situ stress evolution during growth was mea-sured with a multibeam optical stress sensor �MOSS�.25

MOSS data obtained at three different growth rates arepresented in Fig. 3. To obtain these plots, the measured cur-vature, �, was converted to the stress thickness in the film viathe Stoney equation: ���h=�MShS

2 /6. This standard interpre-tation is based on the premise that the film and substrateundergo only elastic deformation, and that the film is con-tinuous. The latter assertion is only a reasonable approxima-tion after the islands have coalesced. For the AlN films stud-ied here this occurs before h reaches 100 nm. Even thoughbending of the substrate-film system is used to measure thestress, the standard Stoney analysis shows that the small

FIG. 2. SEM image of AlN film grown at 750 °C and u=120 nm/h.

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amount of bending that occurs with an unconstrained sub-strate produces only small changes in the film strain �lessthan 1% for all conditions studied�.19

To better understand stress evolution as the film grows,note that ���h=0

h��z�dz, where ��� is the average in-plane

stress over the thickness of the film, and ��z� is the averagein-plane stress at a given distance z from the film/substrateinterface. An important observation in the AIN films consid-ered here is that the measured stress does not change whenthe growth flux is turned off and the specimens are held at orabove the growth temperature for several hours or more.23

This implies that the stress is not altered by relaxationmechanisms �e.g., grain growth, grain-boundary diffusion,etc�. In this situation, the value of ��h� can be obtained di-rectly from the data by differentiating ���h with respect to hto give ��h�=d����h� /dh. Thus ��h� is equal to the slope ofthe data in Fig. 3. The linear portions of the ���h vs h data inFig. 3 then indicate that ��h� has reached a constant value�i.e., d� /dh=0�. This steady-state condition implies that thematerial grown at this point has the same stress as the under-lying crystal �surface stresses will also exist, however, theseshould not affect the stress of a given layer once it has beencovered by subsequent growth�. When ���h vs h exhibits apositive or negative curvature, there is an increase or de-crease in ��h� �i.e., because d� /dh is either positive or nega-tive, such that the material being deposited is either moretensile or more compressive than the material immediatelybelow it�.

The data collected at 90 nm/h in Fig. 3�a� are typical ofmany polycrystalline films. As noted in Sec. I, the character-istic features observed here are a relatively small compres-sive stress prior to island coalescence, followed by tensilestress during and immediately after island coalescence, witha return to compressive stress at longer times. The films inFigs. 3�b� and 3�c� were deposited at higher growth rates,and they exhibit only postcoalescence tensile stress. In par-ticular, the significant stress increase in Fig. 3�c� when thegrowth rate is returned to 300 nm/h demonstrates that a ten-sile mechanism is operating long after island coalescence iscomplete. Thus it is clear that tensile stress evolution is notlimited to the initial contact between neighboring islands.The observation that the postcoalescence steady-state stresscan be either compressive or tensile indicates that bothmechanisms are operating during the growth of continuousfilms.

Another key observation in Fig. 3 is that the steady-statestress values become more tensile as the growth rate in-creases. This cannot be explained with the existing energeticmodels of stress evolution �i.e., with Eqs. �2� and �3��. Evenwhen only tensile stress occurs, the changes in the instanta-neous stress with the growth rate are not adequately de-scribed with the scaling in Eq. �2�. For example, fitting thestress change in Fig. 3�b� with an exponent of B= 1

2 impliesthat �−�� /L� would have to increase by a factor of 6.5 �alarger exponent of B= 2

3 implies an even larger increase in�−�� /L��. Variations in �� of this magnitude are highly un-likely and SEM and AFM measurements show that growthrate changes produce relatively modest variations in the mi-crostructure, with grain sizes for all specimens of 40–80 nmand surface rms roughness of 25–40 nm. Since it is difficultto reconcile the growth rate effects in Fig. 3 with existingenergetic models, we believe that the observed intrinsicstresses are more probably caused by a kinetic competitionbetween the tensile and compressive mechanisms.

FIG. 3. MOSS results for films grown at 750 °C, with different growth rates�due to changes in the Al flux�. �a� Fixed conditions, u=90 nm/h. �b� Initialgrowth at u=120 nm/h, followed by segments with u=300 and 120 nm/h.�c� Initial growth at u=300 nm/h, followed by segments with u=120 and300 nm/h.

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III. KINETIC MODEL

A. Strain generation at grain boundaries

The model developed below describes strain evolution�see Eq. �4�, etc.�. Comparisons between these models andstress data are discussed further in Sec. IV. We represent acontinuous polycrystalline films with a periodic array of is-lands, each of which occupies an identical square footprinton the substrate. The top surface of the film is assumed to becusped, similar to the schematic in Fig. 1�b�. As material isadded to the growth surface, the surface profile and grainsize L in Fig. 1�b� are assumed to be constant. This geometryis adopted because grain-boundary grooving is commonlyobserved in polycrystalline films �although this grooving canalso involve fully or partially faceted surfaces�. With three-dimensional grains, the type of surface profile in Fig. 1�b�can be viewed as either h�y� for a fixed value of x �i.e.,surface position versus y for fixed x�, or as h�x� for a fixedvalue of y. However, the top surface of each grain cannot berepresented by a single profile. For example, even if the di-hedral angle at the cusp is constant along each grain bound-ary, the h�y� profile will vary with the x coordinate �i.e.,because the surface profiles in both the x and y directionsmust be consistent�.

The idealized grain boundaries in Fig. 1�b� are perpen-dicular to the substrate surface, and each grain consists oflayers of atoms that are parallel to the substrate. In derivingthe model, we assume that the strain in the film is determinedby mechanisms that operate at the grain boundaries. This is areasonable assumption for the tensile mechanism, wheregrain-boundary energetics are an inherent feature of previousmodels �see Sec. I and references cited therein�. This is fur-ther supported by the fact that tensile growth strains are notgenerally observed during the growth of monocrystalline,homoepitaxial films of the same or similar materials. Asnoted in Sec. I, there is more uncertainty about the cause ofpostcoalescence compressive behavior. In metal films thereis some experimental evidence which supports the hypoth-esis that a grain-boundary mechanism contributes to postcoa-lescence compressive strain generation.26 However, other re-cent evidence suggests that single-crystal films exhibitcompressive stress behavior which is very similar to poly-crystalline films of the same material.27 This type of investi-gation has not been conducted with the so-called low mobil-ity materials, and for our present purposes we assume thatthe compressive mechanism occurs at the grain boundaries.

With the model microstructure described above, an in-crease in the biaxial tensile strain occurs when a given layerhas fewer atoms than the layer below it. Similarly a layerwhich has more atoms than the underlying layer will shift thestrain in the compressive direction. These strains are a directresult of the constraint imposed by the substrate �i.e., withoutthis constraint, an increase in the number of atoms per layerwould cause an in-plane expansion of the film volume, in-stead of inducing compressive strain�. The average biaxialstrain at a given vertical position, z, can be described by

�z� = XX = YY = −��1 − �Vb�NE

2aL2 , �4�

where a is the height of one layer of atoms, and NE is thenumber of extra atoms in the layer relative to a zero strainconfiguration �i.e., NE=0 corresponds to zero strain, andNE0 and NE�0 lead to compressive and tensile strainsrespectively�. Note that Eq. �4� is derived on a per grainbasis, where each layer contains aL2 /� atoms. The value of�Vb accounts for the changes in the local atomic volume�i.e., volume per site� at the grain boundary, when atoms areeither removed or inserted. If �Vb=0 then there is no volumeadjustment, and �Vb=1 is an upper bound where the missingor additional volume is completely compensated without anyincrease in strain. Values of �Vb between these two limits areexpected for most films, although other values may be pos-sible in some systems. In the context of Eq. �4�, �Vb repre-sents an average value for a wide variety of different atomicconfigurations associated with both the tensile and compres-sive stress mechanisms described below.

The assumption that the dominant intrinsic mechanismsare operating at the grain boundaries, combined with the ab-sence of grain-boundary diffusion, means that the value ofNE in Eq. �4� is determined by the phenomena that occur atthe tip of the cusps in Fig. 1 �i.e., at z=h�. Analysis of theseprocesses is complicated by the stress concentration at thecusp, since this means that the average strain �z� will differfrom the strain at the cusp. However, during the growth ofcontinuous films this discrepancy becomes insignificant atdepths sufficiently far below the cusp, at which point Eq. �4�becomes accurate. Since the cusp moves up as material isdeposited, Eq. �4� will be accurate for a given layer when itis buried underneath the cusp after sufficient additional ma-terial is deposited. In spite of this change in the local strainas growth proceeds �i.e., at a given value of z when it is closeto z=h�, the value of NE is still determined by mechanismsthat operate at the cusp �i.e., when the material in question isfirst deposited�. To obtain an estimate of the distance overwhich the stress concentration at the cusp decreases, considerthe sharp cusp geometry analyzed by Chiu and Gao �Ak=1case in Ref. 20�. With a grain size of 50 nm, this analysispredicts that the stress is within 10% of the average valuewhen the distance below the cusp exceeds 4.3 nm. This cor-responds to less than ten AlN unit cells, which suggests thatthe stress concentration effect described above diminishesquickly as additional growth occurs �note also that cuspswhich are less sharp than those considered by Chiu and Gaowill lead to a less pronounced stress concentration�.

B. Tensile mechanism

Our previous FEM of tensile stress evolution duringpostcoalescence growth applies the energy criteria in Eq. �1�continuously during growth.10 For each time increment inthis model a small amount of material is added to the top ofthe film and the energy minimization problem is then solved.If the small amount of additional material is reduced toatomic-scale dimensions ��1 nm�, the additional grain-boundary closure associated with Eq. �1� is much less thanan atomic dimension �e.g., adding material that is equivalent

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to one atomic layer leads to �b of 0.01 nm or less�.10 Thiscontinuum result suggests that the Nix-Clemens “zipping”mechanism cannot be interpreted at the atomic scale withoutfurther consideration of the physical mechanisms that pro-duce tensile strain during growth. One possibility is that alarge number of atom layers must be added before a zippingincrement on the order of one atomic layer can occur. An-other possibility is that a small number of grain-boundarysites are removed for each atom layer that is added duringgrowth. This latter case is depicted in Fig. 4 for the case ofstep-flow growth.

The mechanism in Fig. 4 provides a generic descriptionof atomic-scale processes that can occur during grain-boundary formation, where the atom positions in each layercorrespond to a simple triangular lattice rather than the actualAlN crystal structure �i.e., we do not differentiate betweendifferent types of atoms, etc.�. This simple model also as-sumes that the the different crystallographic orientations ofthe neighboring grains are due solely to rotational misorien-tations around the z axis. Figure 4�a� is based on the premisethat an incremental segment of grain boundary is created as agrowth step on grain 2 moves along the edge of grain 1 �theheight of grain 1 is at least one atom layer taller�. The keyatom position in this schematic is located in grain 2, imme-diately in front of the step intersection with the boundary.

Templated epitaxial growth on top of the lower level of grain2 would fill this position; however, it is also possible that theatoms in the upper layer that are immediately in front of thestep–grain-boundary intersection will pull together to createa bond, before the growth process can add another atom ontop of this lattice position in the lower layer. When this oc-curs, it will reduce the number of atoms in this layer and leadto an incremental increase in tensile strain.

In some sense, the creation of a missing atom position inFig. 4�b� can be viewed as the atomic scale equivalent of theNix-Clemens “zipping” mechanism. However, this additionalbonding pulls material together along the steps, and is thusoccurring parallel to the substrate surface. In the Nix-Clemens continuum model the additional closure of neigh-boring surfaces is perpendicular to the substrate �i.e., in the zdirection in Fig 1�. In Fig. 4, the local driving force forgenerating additional tension is the energy reduction due tobond formation immediately in front of the step–grain-boundary intersection. Every time an atom position along thegrain boundary is skipped in this way, the top layer containsone atom less than the underlying layer. For each atom layerthat is added, the number of lattice sites removed by thisprocess can then be described by

�NET = −

2La

�S� , �5�

where �S is the area per atom along the grain boundary. Thequantity � is the probability per grain-boundary site that amissing atom occurs �e.g., �=0.01 indicates that 1 grainboundary atom position out of 100 is missing�. Each side ofthe grain boundary contains La /�S interface sites per layer,and the factor of 2 in the numerator accounts for the averagenumber of boundaries where a missing atom can occur �i.e.,four boundaries per grain, each shared by two grains�. ThusEq. �5� describes the change in the number of lattice sites dueto the mechanism in Fig. 4, where the superscript T meansthat Eq. �5� only considers the tensile contribution to �NE. Inaccordance with Eq. �4�, this change in the number of latticesites produces a corresponding change in the strain. If �=0,then �NE

T =0 and the strain in the layer matches the strain inthe underlying layer �assuming that there are no othermechanisms which change NE�. This occurs because thelayer forms epitaxially on the layer below. Note here that thecase where �NE=0 �no change in strain� is considerably dif-ferent from the case where NE=0 �no strain�.

To better understand �, note that the missing atom inFig. 4�b� does not create a lattice vacancy. Instead, a grain-boundary bond is established at the step-grain-boundary in-tersection. A wide range of atomic configurations will pre-sumably occur at these sites, partly because there is a varietyof different atomic arrangements along a given grain bound-ary. The set of different atomic configurations will be furtherexpanded because the grain boundaries in a random poly-crystalline material have a variety of different orientations,and also because of crystallographic and kinetic aspects ofgrowth that affect step orientations. Another consideration isthat certain grain-boundary orientations are more energeti-cally favorable. This suggests that values of � should behigher when a missing atom moves the grain-boundary struc-

FIG. 4. Mechanism for inducing tensile strain when an atomic layer ongrain 2 grows into a neighboring crystallites �grain 1�: �a� partially completelayer in contact with neighboring grain, and �b� top view of the schematic inpart �a� showing how an atomic position can be overgrown, thus producingadditional strain.

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ture towards these lower-energy configurations �and viceversa�. With this in mind, � in our model is an average overof a large assortment of different atomic configurations at thegrain boundaries. Detailed treatments of the atomic-scaleprocesses associated with � are beyond our current scope.However, it is important to note that the tensile stress in thefilm opposes the attractive interatomic force that leads to themissing atom in Fig. 4. This implies that � will decrease asthe tensile stress in the film increases, and that there shouldbe some relationship between � and the driving force fortensile stress creation, ��+Gb. In particular, there is no driv-ing force to create additional tensile stress when the filmstress reaches the limit, ��+Gb=0. To accomplish this in thecontext of the current model, we propose the following phe-nomenological relationship:

� = �o �1 − exp���S/kBT���� + Gb����1 − exp��S��/kBT��

, �6�

where �o is the value of � when Gb=0. Recalling that ���0, note that � decreases towards zero with increasing ten-sile stress, and that �=0 at the limit ��+Gb=0. In this waythe form of Eq. �6� is consistent with an upper limit on thetensile strain in accordance with Eq. �1�.

Although Eq. �6� is not based on specific atomic-scaleconsiderations, it limits the tensile stress in a physically rea-sonable fashion. For cases where the film is far away fromthe tensile limit, � is essentially equal to �o. Then � ap-proaches zero as the stress approaches the limit �i.e., as ��+Gb approaches zero�. In general, the value of �o is dictatedby the atomic-scale details that have been neglected here. Forgrowth at constant conditions �o is treated as a constant;however, this quantity should increase as the magnitude of�� increases �i.e., because the probability for skipping anatom position should increase as �� increases�. In applyingEq. �6�, we currently assume that �o is independent of thegrowth rate, u, under the premise that the bond formation inFig. 4�b� occurs instantaneously �i.e., much faster than theaddition of an atom via the growth mechanism�. In caseswhere this assumption is not valid, a more detailed descrip-tion of �o is needed.

C. Competition between tensile and compressivemechanisms

Intrinsic compressive strain is assumed to occur by ada-tom insertion into the top of the grain boundaries, in re-sponse to the adatom supersaturation on the growth surfaceduring deposition.12 For AlN films where grain-boundary dif-fusion is negligible, this mechanism can be described with.13

�NEC =

4�SL CCo

u�sS − e−�*

� , �7�

where C is the jump rate for inserting atoms at the top of theboundary, �=�M /kT, and Co �at./area� is the adatom con-centration in equilibrium with an unstrained single crystal �aconcentration of Co corresponds to sS=1, and concentrationsthat exceed Co during growth lead to sS1�.13 Following thedescriptions given by Eqs. �4� and �5�, �NE

C is the number ofatoms added by the compressive insertion mechanism, to the

top atom layer of the material that is deposited duringgrowth. The exponential term in Eq. �7� accounts for thelocal strain contribution to the chemical-potential drivingforce to move adatoms from the surface to the top of thegrain boundary. Here the local strain at the top of the grainboundary, *, is distinguished from �Z�, where the latterquantity is the average strain at a distance z into the film �z�h�. As already discussed in connection with the tensilemechanism, cusps at the top of the grain boundary will pro-duce a stress concentration. In this way, Eq. �7� differs fromthe expressions in Ref. 12, where the film surface is assumedto be planar. With cusps, the single values for *, C, and Co

in Eq. �7� are reasonable for a plane strain configuration.However, the key relationship between �z� and NE given byEq. �4� is based on the biaxial strain in the x and y directions,thus suggesting a plane stress configuration. In this case, afull three-dimensional treatment of the cusps would need toconsider variations in *, C, and Co along the top of thecusps. This level of complexity is not needed for our currentattempt to describe stress evolution with respect to the ex-perimental observations reported in Sec. II. Thus, in the con-text of Eq. �7� these quantities are understood to be approxi-mate averages along the cusps.

The tensile and compressive models represented by Eqs.�5� and �7� describe the change in NE that occurs as a singleatom layer is added to the top of the growth surface. This canbe related to the total change in strain according to

d

dh=

�

a=

− ��1 − �Vb�2a2L2 ��NE

T + �NEC� , �8�

where � is the change in strain that occurs as a single atomlayer is added. Note that the second equality in Eq. �8� isobtained by evaluating � with Eq. �4�, and applying �NE

=�NET +�NE

C, where this sum describes the total change inthe number of atoms in the top layer of the film. InsertingEqs. �5� and �7� into Eq. �8� then gives a model that de-scribes the competition between the compressive and tensilemechanisms:

d

dh=

�o

L

�1 − exp���S/kBT���� + Gb����1 − exp��S��/kBT��

−2�TCCo

Lu�sS − e−�*

� . �9�

For simplicity the value of �Vb was assumed to be neg-ligible and was thus set to zero in obtaining Eq. �9�. Kineticmodels typically contain a time derivative which is absent inEq. �9�. This form can be obtained by multiplying both sidesof Eq. �9� by u=dh /dt. This reflects the fact that the time-dependent evolution of the strain is inherently linked to themotion of the growth surface. Note also that the grain aspectratio, h /L, can add a so-called weighted mean curvature con-tribution to the driving forces associated with Eq. �9�.13 Thiseffect accounts for the fact that h /L will, in general, notmatch the equilibrium crystal shape determined by the sur-face and grain-boundary free energies. The weighted meancurvature contribution was neglected in deriving Eq. �9�, be-

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cause the increase in h /L during the growth of our AlN filmsis predicted to shift the steady-state stresses by only a fewmegapascal or less.13

IV. COMPARISONS WITH EXPERIMENTS

In contrast with the strain evolution model developed inSec. III, the experimental results in Sec. II are presented interms of stress, primarily because the Stoney equation showsthat stress is proportional to the measured curvature �withoutthe need to know the film modulus, M�. Our data correspondto the conditions where the Stoney equation is sufficientlyaccurate �i.e., elastic behavior and continuous films of suffi-cient thickness, as noted in Sec. II�. Under these conditionsthe average stress is equal to the product of the average strainand M, for a film with a top surface which is flat and alwaysparallel to the substrate. Freund and Chason show that thisstandard linear relationship is not valid at the point whereinitial contact between islands first occurs and the surface isrough.7 However, the linear relationship should be a reason-able approximation when the film thickness is much largerthan the surface roughness. For this reason, we restrict ourcomparison with the experiments to steady-state stress valuesthat were obtained with film thicknesses that are an order ofmagnitude or more larger than the measured surface rough-ness. The linear regimes observed in Fig. 3 should then cor-respond to steady-state solutions that can be obtained by set-ting Eq. �9� to zero. This condition gives

��1 − exp���S/kBT���� + Gb���

�1 − exp��S��/kBT��= sS − exp�− �*� ,

�10�

� ��ou

� CCa .

Here the driving forces for tensile and compressive stressgeneration are described by �� and sS, respectively. Therelative rates of the competing kinetic mechanisms are incor-porated into the parameter � �keeping in mind that �o alsodepends on �� in a way that is not rigorously defined heredue to our current lack of detailed knowledge about thisbehavior�.

If the local strain at the cusp, *, is far below the maxi-mum tensile strain, then the left-hand side of Eq. �10� isapproximately equal to �. In this case the existing strain inthe film has a minimal effect on the tensile mechanism de-scribed by Eq. �6� �i.e., ��o�, and Eq. �10� can be rear-ranged to give

* � −1

�ln�sS − �� . �11�

Based on Eq. �11�, * approaches the energetic limit for thecompressive mechanism as �→0 �i.e., Eq. �3��, and the tran-sition from compressive to tensile steady-state strain occursat ���sS−1� �i.e., the steady-state strain is tensile for �values above �sS−1��. Thus for films far away from the ten-sile limit, Eq. �11� predicts that the kinetic parameter �modifies the compressive driving force �i.e., sS� in a linearfashion that shifts the steady-state strain in the tensiledirection.

When � is large, Eq. �11� is not valid and we must revertto Eq. �10� where the right-hand side vanishes as �→ +�.This limit corresponds to the criteria described by Eq. �1�.Thus an important result from Eq. �10� is that values of �between 0 and +� correspond to steady-state stress valuesbetween the compressive and tensile limits. A similar dimen-sionless group was proposed in an earlier paper, where thetensile mechanism was not specifically addressed.12

More detailed consideration of Eq. �10� requires a quan-titative description of Gb. While Gb depends on the actualcusp geometry at the top of the grain boundary, we can ob-tain insight into the behavior of Eq. �10� by considering thecycloid surface cusps that were used in the Nix-Clemensmodel, where Gb= ���2L /2E.6 These cracklike cusps exhibita stress singularity, and Chiu and Gao’s analysis suggests anapproximation of the form *�� L /4�a.20 In this expres-sion, the atomic dimension a is used as an approximatecrack-tip dimension, with the understanding that a more re-alistic description of * requires consideration of atomic-scale phenomena. We then use values of �=665, L /a=207,and �S /kBT=5.93 m2/J that are obtained from basic infor-mation about the AlN films in Sec. II, to obtain the plots inFig. 5. The form of Eq. �11� suggests that these �* vs ��−sS� curves are a convenient way to examine steady-statebehavior. With this in mind each of the curves in Fig. 5�a�was obtained from Eq. �10� by varying � with sS constant,and plotting the results as a function of ��−sS�. All of thesecurves converge at sufficiently low values of ��−sS�, in ac-cordance with Eq. �11�. As ��−sS�→−0, Eq. �11� is nolonger valid and there is a sharp increase in tensile stress. Inthis region the use of different sS values produces somevariation in the results; however, as �→ +� the effects ofvarying sS diminish and all of the solutions converge to thesame limiting value, which for cycloid surface cusps is givenby �*→ �−��E�2� / �2�a�kBT�2� �this limit is not fullyapparent in Fig. 5�a�, because solutions for large enough �are not shown�.

The effect of �� on the limiting value of �* as �→+� can be seen in Fig. 5�b�. These plots show that �� alsohas a significant effect on the sharp transition that occurs as��−sS�→−0. In general, the steady-state solutions are moresensitive to changes in �� �Fig. 5�b�� than they are tochanges in sS �Fig. 5�a��. This sensitivity to �� also impliesthat Gb will have a significant impact on �* in the regimewhere ���+Gb� is small enough to make the exponential onthe left-hand side of Eq. �10� significant �i.e., as the stressapproaches the continuum tensile limit�. Thus changing thefunctional form of Gb will also alter the value of ���+Gb�,and have a significant impact on �* when � is large enough.

The key experimental observation in Fig. 3 that contra-dicts previous energetic models is that higher growth ratesshift the steady-state stress to higher tensile values. However,this result is consistent with Eq. �10�. To see this, note thatFig. 5 shows that higher tensile strains are predicted forhigher values of ��−sS�. To understand how Fig. 5 compareswith experiments, note that Eq. �10� indicates the higher ��−sS� values should correspond to higher growth rates, basedon the premise that � is proportional to u while sS is ex-pected to be a relatively weak function of u. To compare Eq.

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�10� and the data, consider the results in Figs. 3�b� and 3�c�where there are several different transitions between growthrates of 120 and 300 nm/h. Using the cycloid cusp modelagain, the measured steady-state values from Fig. 3 wereconverted to the �* values plotted in Fig. 5�b�, where thefilled and open circles correspond to 120- and 300-nm/hgrowth rates, respectively. These �* values were then fittedto the ��=−0.2 J /m2 plot �i.e., Eq. �10��, to obtain values of��−sS� for each of the two growth rates.

Fitting the data in Fig. 5�b� leads directly to values of��−sS�, without making any assumptions about the way inwhich the growth rate affects the values of � and sS. Intheory, the ��−sS� values that are obtained in this way can beused to obtain quantitative information about � and sS. Todemonstrate this, Eq. �10� was fitted directly to a larger set ofdata by assuming that � is proportional to u �per Eq. �10��and that sS is independent of u. The corresponding compari-son in Fig. 6 was obtained by using ��=−0.2 J /m2 and sS

=1.5, which gives � /u=0.0123 nm/h. The circles in thisplot correspond to the data in Fig. 5�b� and additional datapoints from other experiments are denoted with diamonds. Alarger value of �� is probably more realistic, however, in-

creasing the magnitude of �� gives a poorer fit to the data.One justification for a low value of �� is that the cycloidsurface used to evaluate Gb considers cusps that are sharperthan those on the actual growth surface. To some extent thisdiscrepancy can be compensated with an artificially lowvalue of ��, which may provide some explanation for theresults of our fitting procedure.

In applying Eq. �10� it is important to note that � and sS

may vary with u in ways that are more complex than thesimple relationships that were used to obtain Fig. 6. For ex-ample, increasing u can increase the adatom supersaturationsS. The relationships between surface supersaturation andcrystal growth have been studied in detail for many decades,and there is a considerable literature that considers theseprocesses.28 More detailed analysis of these effects and theirrelationship to the current model is of interest, and is cur-rently being pursued. With the proposed compressive mecha-nism, Eqs. �7�–�10� predict that an increase in sS due to fastergrowth should shift the intrinsic stress in the compressivedirection, which contradicts the experimental observations.This can be explained by noting that the growth rate has alarger effect on �. The explicit relationship between � and uin Eq. �10� reflects this effect. Note also that � dependsdirectly on �o, the probability associated with tensile stressevolution �see Eq. �6��. This quantity may also depend on u,if the addition of an atom via growth competes with theproposed mechanism for tensile stress generation. However,for current purposes we assume that this bond formation isfast relative to growth. If this process is not instantaneousthen the relationship between � and u might be somewhatless than first order. Based on all of the above considerations,it is likely that both � and sS increase with u, although theexact nature of these effects is potentially complex and notwell understood at this time. Note that the experimental ob-servation that faster growth shifts the stress in the tensiledirection is consistent with Eq. �10� as long as � increaseswith u at a faster rate than sS increases with u �i.e., since thisimplies that ��−sS� will increase with u�. In terms of the fitin Fig. 6, differences in the way that � and sS vary with uwill change the solid line that was obtained with Eq. �10�.

It is encouraging that the model in Eq. �10� is generally

FIG. 5. Plots of Eq. �9�, for �a� different values of sS, with a fixed value of��=−1.0 J /m2. �b� Different values of ��, with a fixed value of sS=1.5.The filled and open circles in Fig. 5�b� correspond to the data obtained atgrowth rates of 120 and 300 nm/h, respectively.

FIG. 6. Stress vs growth rate. The circles correspond to the data pointsshown in Fig. 5�b�, and the diamonds correspond to the data points fromother growth runs. The curve was fitted to the data with Eq. �9�, using sS

=1.5 ��� ��=−0.2 J /m2.

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consistent with the observed growth rate effects. However,the limited number of data points and the amount of scatterseen in Fig. 6 imply that this model should be investigated inmore detail. In terms of the model, uncertainties associatedwith evaluating Gb, ��, �, and sS are discussed in the pre-ceding paragraphs. In terms of the data, we believe that theobserved scatter in Fig. 6 is at least partly due to variations inthe microstructures of different films. For example, thesteady-state stresses obtained from Figs. 3�b� and 3�c� atgrowth rates of 120 and 300 nm/h show similar trends; how-ever, the actual stress values are not identical. The differentgrowth rate sequences in these two films lead to some differ-ences in the film microstructure that may be responsible forthe relatively modest differences in the steady-state stressvalues.

The existing data do not provide sufficient evidence toprove or disprove the specific tensile and compressivemechanisms that are used to formulate Eqs. �9� and �10�. Asnoted above, the key result which is consistent with our ex-periments is the predicted growth rate dependence. However,the competition between other tensile and compressivemechanisms can lead to similar growth rate effects. For ex-ample, a growth surface that is rougher than that depicted inFig. 4 could lead to the creation of very small voids at thegrain boundaries, with tensile stress resulting from the col-lapse of these voids.29 Missing atom positions similar to theone depicted in Fig. 4�b� could also occur if excess vacanciesin the bulk grain diffuse to the grain boundaries, althoughthis requires bulk diffusion which is expected to be very slowin AlN �this vacancy annihilation mechanism might be morerealistic in materials with higher vacancy mobilities�. An-other possible compressive mechanism is Spaepen’s kinetictrapping of atoms in the growth surface.14 A quantitative de-scription of this phenomena has not been developed, how-ever, one approach is to assume that it is also driven by theadatom supersaturation that leads to the grain-boundary in-sertion mechanism in Eq. �7�. In this case, the introduction ofcompressive strain can be described with an expression simi-lar to Eq. �7�, except that it does not depend on the grainsize. In principle any of these alternative models can be rep-resented with a simple rate equation analogous to Eq. �9�,based on appropriate descriptions of how NE varies �see Eq.�4�, etc.�. The central idea here is to describe the competitionbetween different tensile and compressive mechanisms.From our current work, it appears that detailed study of bothgrain size and growth rate effects can provide a much betterbasis for evaluating specific mechanisms.

V. CONCLUSIONS

The AlN experiments reported here demonstrate that theintrinsic stress is a strong function of the film growth rate.These data are not consistent with previously proposed mod-els of stress evolution. Thus a model that describes the ki-netic competition between tensile and compressive stressmechanisms is proposed. This competition is quantified witha single parameter, �, as seen in Eq. �10�. The �→0 limiting

case corresponds to our previous energetic model of a com-pressive stress determined by the adatom supersaturation�i.e., Eq. �3��, while the �→ +� limiting case corresponds toa tensile stress limited by the strain energy release rate at thetop of the grain boundary �i.e., Eq. �1��. The observed AlNresults appear to fall between these two limits. The proposedmodel will clearly benefit from further refinement. In par-ticular, a more accurate understanding of atomic-scale phe-nomena that occur at the top of the grain boundary is impor-tant. While our model is formulated with specific tensile andcompressive mechanisms that are associated with grainboundaries, the general approach that is used here can alsobe readily extended to describe other intrinsic stress mecha-nisms.

ACKNOWLEDGMENT

Research support from the National Science Foundationunder Award Nos. DMR-0079964 and DMR-0305418 isgratefully acknowledged.

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