Some Practical Issues of Curvature and Thermal …mingdao/papers/2008_J.Elec.MAter...Some Practical Issues of Curvature and Thermal Stress in Realistic Multilevel Metal Interconnect

Some Practical Issues of Curvature and Thermal Stressin Realistic Multilevel Metal Interconnect Structures

T.-S. PARK,1,4 M. DAO,1 S. SURESH,1 A.J. ROSAKIS,2 D. PANTUSO,3

and S. SHANKAR3

1.—Department of Materials Science and Engineering, Massachusetts Institute of Technology,Cambridge, MA 02139, USA. 2.—Graduate Aeronautical Laboratories, California Institute ofTechnology, Pasadena, CA 91125, USA. 3.—Technology CAD Group, Intel Corporation, Hillsboro,OR 97124, USA. 4.—e-mail: [email protected]

This paper presents the results of a systematic study of curvature and stressevolution during thermal loading in single- and multilevel interconnect linestructures which have been deposited on a much thicker substrate. Effects ofline aspect ratio, passivation geometry, and metal density within a metal-ization level on thermal stress evolution in the lines are addressed. The cur-rent analytical stress model enables us to predict that interaction betweenlines on the same level, i.e., in the lateral direction, is so strong that it cannotbe neglected. A two-dimensional (2-D) finite element method has been used toverify the accuracy of the current model, while available experimental datahave been compared with theory. In order to capture the exact variation of thethermal stresses at different metalization levels, and to investigate the effectof the upper level line arrangements on the stress states at the lower level, athree-dimensional (3-D) finite element analysis was employed. It can be seenthat the interaction between levels in the vertical direction is quite weak whenthe thickness of the interlevel dielectric (ILD) layer becomes comparable tothat of the metal layer.

Key words: Copper, curvature, multilevel metalization, thermal stress,thin-film structure

INTRODUCTION

It is desirable to evaluate stresses in interconnectlines formed on a Si substrate to improve the designof the device structure, selection of materials, fab-rication processes, and other aspects of the device toenhance its manufacturability and reliability.Stress measurements can be used to facilitatequality control during various fabrication stepsincluding film deposition, patterning, polishing, andthermal cycling due to repeated annealing andcuring steps. Determining stress values in the linesalso enables us to assess the reliability of theinterconnect materials against failure from suchphenomena as electromigration,1–3 stress-inducedvoiding,4,5 and hillock formation.6 As new, low-modulus, high-porosity insulating materials are

introduced because of their low dielectric constants,additional failure mechanisms, such as yielding andcracking in the dielectric layer embedding the metallines and in the interface between the two, are alsoobserved.7

The stresses in the interconnect structure mayresult from various sources. One of the major factorsaffecting reliability stems from stresses produced bythermal mismatch between metal lines and sur-rounding materials. Various experiments based onX-rays8 and substrate curvature measurements9–11

have been performed for the case of film structurescontaining metal lines embedded within dielectricmaterials. In parallel, finite element analysis (FEA)has been employed to estimate volume-averagedstresses in metal lines.12–14 Although FEA canincorporate detailed information of the metal lines,including inelastic material properties and/orgeometry of barrier layers,14 it usually requireslengthy computations, which must generally be

(Received August 27, 2007; accepted February 12, 2008;published online March 18, 2008)

Journal of ELECTRONIC MATERIALS, Vol. 37, No. 6, 2008

DOI: 10.1007/s11664-008-0409-4� 2008 TMS

777

repeated each time the line geometry is changed. Incontrast, analytical predictions provide very goodinsight into the parameter controlling the defor-mation process, and also provide reasonably accu-rate results in a time-efficient manner.15–17 UsingEshelby�s inclusion theory, Niwa et al.15 modeledthe line as a cylinder of elliptical cross-sectionembedded in an infinite isotropic matrix having thesame elastic properties as the line. This model wasgeneralized by Korhonen et al.16 by using differentelastic properties between the line and the matrixwhile still neglecting the substrate. These earlyanalyses, based on Eshelby�s theory, can incorpo-rate aspect ratios of the lines, but fail to capture theeffects of the substrate and of the neighboring lines,while completely neglecting sharp edges/corners ofthe lines. Wikstrom et al.17 were the first to proposea fully fledged thermoelastic analysis of periodic,alternating, metal/dielectric lines on a substrate.This model considered the two limiting cases, lowand high line aspect ratios, which is similar to theanalysis employed in the present work. Wikstrom�smodel, however, did not account for substratedeformation, and as a result, was unable to providewafer curvature information, which is directlyrelated to the volume-averaged stresses in both themetal and the dielectric lines on the substrate. Thismodel was also strictly developed for a single-levelstructure deposited on a much thicker substrate.

As the industry is pushing for ever-increasinglevels of metalization (a great need for more inte-grated semiconductor chips), the need for multilevelstress analysis tools is also becoming imminent.Nevertheless, very little work has been done, onboth analytical and numerical modeling, of multi-level thin-film/substrate systems. In perhaps thefirst of these studies, Shen18 presented numericalresults on the evolution of thermal stresses in verysimple multilevel structures. Two levels of metallines, aligned vertically or arranged in a staggeredmanner, were modeled using two-dimensional (2-D)FEA. This analysis quantitatively estimated inter-actions between upper and lower metalization lev-els. Due to limitation of a 2-D analysis, it could notproperly take into account a truly three-dimen-sional (3-D), realistic, multilevel line geometryhaving different line directions at each level. Fur-thermore, there was no attempt to relate wafercurvature to feature stress. Such a relation is veryimportant for process control in all stages ofmetalization (e.g., chemical-mechanical polishing(CMP) following each metalization step).19

In the present work, wafer curvatures as well asvolume-averaged stresses in lines are calculated foralternating metal/dielectric lines on a substrateduring temperature changes. This incorporates thestress-curvature relation and additional conditionsrelevant to limiting line geometry. Two limitingcases are considered. The first is relevant when lineaspect ratios are very low. In this case, each line canbe considered an individual film. In contrast, when

the aspect ratios are sufficiently high, the metal anddielectric lines can be modeled as a homogeneousbut anisotropic composite layer with differenteffective thermoelastic properties along and acrossthe lines. Numerical analysis using 2-D FEA hasbeen employed to evaluate the validity of the theo-retical predictions for these two limiting cases.Furthermore, 3-D FEA was carried out for thepurpose of capturing the realistic, three-dimen-sional nature of multilevel structures and ofexploring the possibility of extending the analysis tomore complicated line geometries. Finally, availableexperimental results have been compared with bothanalytical and numerical predictions.

The ultimate purpose of this paper is a practicalone. Its goal is to demonstrate that analysis of thetype described in the above can indeed be appro-priately generalized to cover realistic multilevel filmstructures of practical interest in integrated circuit(IC) manufacturing. The detailed comparisons ofanalytically computed stresses to the 2-D and the3-D numerical simulations establish the limitationsof the purely closed-form, but approximate, analyt-ical approach. They also establish the range of itsapplicability to practical stress problems in morerealistic structures.

ANALYTICAL MODEL

Single-Level Structure

Low Line Aspect Ratio

When metal and dielectric lines are very short andwide (Fig. 1a) each line can be treated as an indi-vidual thin layer (Fig. 1b).17 Since the lines are verywide across the line direction, and the line height isvery small compared to both its length and width,the stress state in the lines is basically that of equ-ibiaxial plane stress. Therefore, the volume-aver-aged stress values along the line (x-direction), andacross the line (y-direction) are the same, as are thecurvatures in the x- and y-directions, for this limit-ing line geometry. From force and moment balanceequations of the metal thin-layer structure,20,21 thevolume-averaged stresses from thermal mismatch inthe metal line, hrli, and the curvature changes of themetal/substrate structure, Djl, during a tempera-ture change, DT, can be expressed as,

hrlxxi ¼ hrl

yyi ¼ �El al � asð ÞDT; hrlzzi ¼ 0; (1)

Djlx ¼ Djl

y ¼6

Es

h

h2s

Elðas � alÞDT; (2)

where El and al are the biaxial modulus (El/[1 - ml])and thermal expansion coefficient of the thin metallayer, respectively, Es and as are the correspondingproperties of the substrate, and h and hs are thethickness of the layer and the substrate, respec-tively. Similar expressions can be obtained for the

Park, Dao, Suresh, Rosakis, Pantuso, and Shankar778

volume-averaged stresses in the dielectric line, hroi,and the curvature changes of the dielectric/sub-strate structure, Djo, with corresponding thermo-elastic properties of the dielectric lines. The overallcurvature changes of both the metal and dielectriclayers on the substrate (Fig. 1a) are simply the sumof contributions from each layer (Fig. 1b).

High Line Aspect Ratio

Figure 2a is a schematic of metal interconnect lines(e.g., copper) with dielectric lines (e.g., oxide) on a Sisubstrate following the Damascene process. Assum-ing that the aspect ratio (h/w) of the metal and

dielectric lines is sufficiently high, the structure canbe homogenized into a composite layer as shown inFig. 2b.17,22 Due to the anisotropic line geometry, thiscomposite layer has different values of effectiveelastic modulus and thermal expansion coefficientalong the lines, Ex and ax, respectively, from thoseacross the lines, Ey and ay, respectively, even if thematerial itself is modeled as elastically and thermallyisotropic. Using composite theory, these effectiveproperties can be calculated in terms of the volumefraction, elastic modulus, and thermal expansioncoefficient of the metal lines, fl (= w/p), El and al, andthose of dielectric lines, fo (= 1 - w/p), Eo, and ao.

22

Fig. 1. Schematic of Cu interconnect lines without a passivation layer on a Si substrate following the Damascene process at low line aspect ratio.

Fig. 2. Schematic of Cu interconnect lines without a passivation layer on a Si substrate following the Damascene process at high line aspectratio.

Some Practical Issues of Curvature and Thermal Stress in Realistic MultilevelMetal Interconnect Structures

779

Park and Suresh22 proposed a simple uniaxial,anisotropic composite model for patterned elasticlines to predict curvature evolution in response tochanges in internal stresses arising from thermalcycling. Their model provides reasonably accuratevalues compared to detailed finite element analysis.The curvature changes of the wafer along andacross the lines, Djx and Djy, respectively, during atemperature change, DT, are expressed as,

Djx ¼6

Es

h

h2s

Exðas � axÞDT; (3)

Djy ¼6

Es

h

h2s

Eyðas � ayÞDT: (4)

By convention, a positive curvature changeresults in a concave shape of the line structure(composite layer) side of the substrate. When thelayer is much thinner than the substrate, the vol-ume-averaged normal stresses in the entire layeralong and across the lines, hrxxi and hryyi, respec-tively, can be obtained from curvature changesduring a thermal excursion using an anisotropicform of the well-known Stoney formula.23,24 Sincethe thickness of the layer is much less than that ofthe substrate, the net out-of-plane average stress,hrzzi, in the layer vanishes. In the limit of highaspect ratio, the average normal strains on the x-zplane for both metal and dielectric lines should bethe same (isostrain condition; h�lxxi ¼ h�oxxi andh�lzzi ¼ h�ozzi).

17,25 In addition, equilibrium consider-ation shows that the averaged stresses in they-direction for both lines must be equal (isostresscondition; hrl

yyi ¼ hroyyi).

17,25

By employing the above conditions, the volume-averaged stresses in the metal and dielectric linescan be computed from the following matrix equa-tion:

fl 0 0 fo 0 0

0 fl 0 0 fo 0

0 0 fl 0 0 fo

1El

� ml

El� ml

El� 1

Eo

mo

Eo

mo

Eo

� ml

El� ml

El

1El

mo

Eo

mo

Eo� 1

Eo

0 1 0 0 �1 0

26666666664

37777777775

hrlxxihrl

yyihrl

zzihro

xxihro

yyihro

zzi

26666666664

37777777775

¼

C Djx þ msDjy

� �

C msDjx þ Djy

� �

0

ao � alð ÞDT

ao � alð ÞDT

0

2666666664

3777777775;

(5)

where C ¼ 16

h2s

hEs

1�m2s

and Djx, Djy are the net substratecurvature changes in the two orthogonal directions,parallel and perpendicular to the lines that are

associated with a change of temperature DT. In thisformula h is the height of the lines within theperiodic structure (without a capping layer).

Since this is for the limiting conditions of a veryhigh aspect ratio, this model does not account forthe effect of line aspect ratio. In addition, at highaspect ratio, each line is contacted on its top in avery limited region with a passivation layer which isdeposited during postprocessing. As a result, thepresent model, which does not take into account acapping layer, is still expected to yield accuratestress predictions irrespective of postprocessing.Indeed, as integration of semiconductor devicesincreases, and the line aspect ratio becomes higher,this theory becomes progressively more practical forcalculation of volume-averaged stresses in inter-connect lines.

Multilevel Structure

The evolution of curvatures of a multilevelstructure can also be assessed using the abovemodel by invoking the concept of superposition.Figure 3 shows a schematic of two trilayer struc-tures with different metal line arrangements on theupper level. One is an aligned structure with lineson the upper level running in the same direction asthose on the lower level (Fig. 3a). The other is aperpendicular structure with lines on the upperlevel running in a perpendicular direction to thelower-level lines (Fig. 3b), which is a more realisticline arrangement in the semiconductor interconnectdesign. In the high-aspect-ratio limit, each struc-ture can be regarded as the superposition of twocomposite layers, with an interlevel dielectric layer(ILD) between them. For a given ILD material, thecurvature change resulting from the ILD layer, Djp,is obtained in terms of the biaxial modulus of theILD layer, Epð¼ Ep=½1� mp�Þ; its thermal expansioncoefficient, ap, and its thickness, hp, as,

Fig. 3. Superimposition of individual metal and ILD layers resultingin a trilayer structure after the dual Damascene process: (a) alignedstructure and (b) perpendicular structure.


Djp ¼6

Es

hp

h2s

Epðas � apÞDT: (6)

By adding Djp from the ILD layer (Eq. 6) to thevalues of Djx and Djy from the two composite layers(Eqs. 3 and 4), respectively, the curvature changesof this trilayer structure, Dj0x and Dj0y; are obtained:

Dj0x ¼ Djxðlower layerÞ þ DjpðILD layerÞþ Djxðupper layerÞ:

(7)

Dj0y ¼ Djyðlower layerÞ þ DjpðILD layerÞþ Djyðupper layerÞ:

(8)

The structure shown in Fig. 3 can be obtained bythe dual Damascene process.19 It may be followed byadditional processes such as another passivationlayer deposition and further metalization. The con-cept of superposition can be applied to structureswith more than three layers provided that linearelastic conditions are maintained.

FINITE ELEMENT METHOD

The finite element method (FEM) was used toverify the present analytical model. For this pur-pose, the finite element program ABAQUS26 wasemployed. Table I shows the material propertiesused in the simulations.13,23 In this work Cu andSiO2 were chosen as the model metal and dielectricmaterials, respectively, in light of the shift fromconventional Al-based interconnects to new CuDamascene structures in the semiconductor indus-try. Isotropic material properties were used andresidual stresses, which may result from Cu depo-sition and/or polishing, were not taken into account.This analysis deals only with elasticity, which isexpected to be valid over a wide range of practicalinterest since the material surrounding the Culines, especially in the case of passivated lines, leadsto elevated levels of hydrostatic stress in the lines,thereby inducing constrained deformation.9

2-D Generalized Plane-Strain Model

A unit cell illustrating the mesh used in the 2-Dnumerical simulation is shown in Fig. 4. Due to the

periodicity and symmetry of the arrangement, onlya unit segment ranging from a symmetry axis andthe neighboring periodic boundary is needed. Hereh, w, and p represent the height, width, and pitch ofthe Cu lines, respectively. At the reference point 0(y = 0, z = 0) in Fig. 4, the displacements along they- and z-directions are taken to be zero. Along thesymmetry axis (y = 0), the displacement in they-direction vanishes. This means

uy ¼ 0 at y ¼ 0: (9)

The outer boundary of the unit cell (y = p/2) is freeto move, but it is forced to remain on a straight linefollowing the assumptions of classical bending the-ory. This gives

y ¼ C1zþ C2 at y ¼ p

2; (10)

where C1 and C2 are constants to be determinedwith further analysis. No other boundaries areconstrained during the thermally induced defor-mation. The wafer curvature change across the lines(in the y-direction) can then be directly calculatedfrom the relative positions of the representativepoints on the boundary, y = p/2;

Fig. 4. A representative unit cell (top portion) and finite elementdiscretization for Cu lines with SiO2 on the single layer for the 2-Dgeneralized plane-strain model.

Table I. Material Properties Used in theSimulations

E (GPa) v a [10-6/�C]

Si 130 0.28 2.60Cu 110 0.30 17.0SiO2 71.4 0.16 0.524Si3N4 380 0.20 2.25


781

Djy ¼2

hspf½uy�y¼p

2; z¼�hs� ½uy�y¼p

2; z¼0g; (11)

where hs is the thickness of the substrate.A generalized plane-strain formulation, which is

an extension of the plane-strain framework (withthe y-z plane being the plane of deformation), isused in the calculations.12,13 This is done bysuperimposing a longitudinal strain, exx, on theplane-strain state. To properly simulate the actualresponse of the patterned parallel lines on thesubstrate, the strain field exx is constrained toinduce a constant rotation about the y-axis,

@2uz

@x2¼ C3; (12)

where C3 is a constant directly determined by theanalysis. The three-dimensional effect can thus beadequately described by the present model. It givesthe wafer curvature change in the x-direction:

Djx ¼@2uz

@x2: (13)

The generalized plane-strain model is thus capableof yielding more realistic field quantities in bothphases than the strict plane-strain formulation.

3-D Model

Representative unit cells with different linearrangements on upper levels used in the 3-Dnumerical simulation are shown in Fig. 5. As in the2-D simulation, the periodicity and symmetryfacilitate use of only a unit segment ranging fromsymmetry planes to the neighboring periodicboundaries for calculation. At the reference point 0(x = y = z = 0), the displacements in all the direc-tions are assigned to be zero. On one symmetryplane (x = 0), the displacement in the x-directionvanishes. On the other symmetry plane (y = 0), allthe nodes are fixed in the y-direction.

The outer boundaries of the unit cells (x = l andy = p/2) are free to move, but they are forced toremain as flat planes. This can be obtained by con-tacting the outer surfaces with frictionless, rigid,and flat planes in the model. The curvature changein the x-direction can then be calculated from the

Fig. 5. Representative unit cells (top portion) for Cu lines with SiO2 on the two level structure after the dual Damascene process for the 3-D finiteelement model: (a) aligned structure and (b) perpendicular structure.


relative positions of the representative points on theboundary, x = l;

Djx ¼1

hslf½ux�x¼l; y¼p

4; z¼�hs� ½ux�x¼l; y¼p

4; z¼0g: (14)

Similarly, the curvature change in the y-directionis obtained from the relative positions of the repre-sentative points on the other boundary, y = p/2;

Djy ¼2

hspf½uy�x¼ l

2; y¼p2; z¼�hs

� ½uy�x¼ l2; y¼p

2; z¼0g: (15)

In order to confirm the validity of the 3-D FEM,the results from passivated lines on the single-levelstructure were compared to those from the 2-Dgeneralized plane-strain model. The values for thestress distribution, as well as volume-averagedstresses and wafer curvature changes, were thesame in both cases. It was thus established that it isindeed possible to use 3-D FEM for multilevelstructure as an extension of 2-D FEM for single-level structure.

RESULTS AND DISCUSSION

Single-Level Structure

Here the current analytical model for stresscomponents on features and overall substrate cur-vature changes is compared with finite elementanalysis using a 2-D generalized plane-strain modelin various line geometries of varying line aspectratio (h/w), passivation ratio (hp/h), and differentcapping materials. This parametric study is done toinvestigate the range of validity and of accuracy ofour simple analytical model and to identify possiblereasons for discrepancies between the theoreticaland the numerical predictions. The effect of metaldensity (fl = w/p) is discussed in the following sec-tion. A comparison with available experimentaldata is also presented.

Curvature Evolution as a Function of Line AspectRatios

Figure 6 shows the ratios of the thermal curva-ture coefficients ðb ¼ Dj=DTÞ between along andacross lines, as a function of aspect ratio, at a fixedpitch ratio (p/w = 2). In the low-aspect-ratio limit,the lines behave like individual Cu and oxide thinfilms. The curvature changes in the x- and y-direc-tions are therefore identical. In the high-aspect-ratio limit, it can be seen that the lines collectivelyconstitute a single composite layer with differenteffective thermoelastic properties, which results inanisotropic curvature values. FEM results are ingood agreement with theoretical values at the highaspect ratio limit even down to h/w = 1, thenapproach the other theoretical limit as the aspectratio becomes lower. Although it is difficult to com-pare the current analysis to absolute experimental

curvature values directly due to the lack of exactthermoelastic property data for Cu and oxide linesin real specimens, comparison of the ratios of cur-vature changes between along and across the linedirections would be useful. Curvature measure-ments of Damascene Cu lines on a Si substrate withdifferent aspect ratios were previously reported.11

These values are reasonably matched with thecurrent analytical and numerical results, withinexperimental scatter, as shown in Fig. 6.

Effect of Line and Passivation Geometry on StressEvolution

Figure 7 illustrates the normal stress componentsduring cooling from 200�C to room temperature(20�C) in Cu lines without passivation (capping)layer as a function of aspect ratio (h/w), at a fixed Cudensity (fl = 0.5). In the low-aspect-ratio limit, theCu lines are so wide that they can be treated as thin,continuous Cu films. Therefore, an equibiaxialplane-stress condition prevails, as shown fromEq. 1. FEM results deviate from the values expectedfrom theory for the intermediate aspect ratios.However, as the aspect ratio increases, these resultsdo approach the theoretical predictions, based onEq. 5. This is due to the assumption of the currenttheory that at the high-aspect-ratio limit individualCu and SiO2 lines are sufficiently tall so that theadditional conditions, i.e., isostrain conditions onthe x-z plane and isostress condition in they-direction, can be used to calculate thermal stres-ses in each line. From Fig. 7 the theoretical valuesat the high-aspect-ratio limit underestimate hrl

yyi;

Fig. 6. Ratios of thermal curvature coefficients between along (x)and across (y) line directions of single-level structure without pas-sivation (w/p = fl = 0.5, h = 1 lm and hs = 525 lm) as a function ofaspect ratio (h/w); low-aspect-ratio limit from Eq. 2 and high-aspect-ratio limit from Eqs. 3 and 4.


783

and overestimate hrlzzi compared to FEM results at

intermediate line aspect ratios. hrlxxi is well matched

over a wide range of aspect ratios. The reason forthese discrepancies between theory and FEM will bediscussed later, along with the effects of a top pas-sivation, or capping layer.

The variations of normal stress components areshown in Fig. 8 as a function of the thickness of theSiO2 passivation layer. The hrl

xxi values are notheavily influenced by top passivation and correlatequite well with the theory, as shown in Fig. 8a.Values of hrl

yyi; however, increase with passivation,which leads to a more pronounced deviation inintermediate aspect ratios from the theoreticalpredictions at the high-aspect-ratio limit (Fig. 8b).Values of hrl

zzi also increase when passivated,resulting in better agreement in intermediate aspectratios with theoretical predictions in the high-aspect-ratio limit, Fig. 8c. For both hrl

yyi and hrlzzi; there is

little difference when the thickness of the passivation

layer increases by a factor of 5.Figure 9 shows contours of ryy in a Cu line and an

oxide trench at various aspect ratios and fixed metaldensity (fl = 0.5) when passivated with a SiO2 layer(hp/h = 1). At h/w = 1, the Cu line generally showshigher stresses than the oxide trench. When linesare taller (h/w = 2), the isostress condition acrosslines seems valid in the middle of lines and the FEMvalues in this region are quite well matched withthe theoretical prediction from Eq. 5. Near inter-faces between underlying/overlying (passivation)layers, however, deviation resulting from thermalmismatch appears. Since SiO2 is chosen for theoverlying layer in this case, localized stresses in theCu line near an interface between the overlyinglayer are higher than stress values in the middle ofthe line. In addition, high localized stress near aninterface between the underlying Si substrate are

found resulting from large thermal mismatchbetween the Cu line and the Si substrate. This iswhy the volume-averaged stresses of Cu lines in they-direction, hrl

yyi; exhibit higher values than the

Fig. 7. Volume-averaged stresses in Cu lines on single-levelstructures without passivation (w/p = fl = 0.5, h = 1 lm and hs =525 lm) as a function of aspect ratio (h/w) during cooling from 200�Cto room temperature; low-aspect-ratio limit from Eq. 1 and high-aspect-ratio limit from Eq. 5.

Fig. 8. Volume-averaged stresses in Cu lines on single-level struc-tures with various passivation thicknesses (w/p = fl = 0.5, h = 1 lmand hs = 525 lm) as a function of aspect ratio (h/w) during coolingfrom 200�C to room temperature: (a) hrl

xx ; i (b) hrlyy i; (c) hrl

zzi; low-aspect-ratio limit from Eq. 1 and high-aspect-ratio limit from Eq. 5.


theory, as shown in Fig. 8b. Localized stresses inthe oxide trench near an interface between theoverlying layer are smaller than the stresses in themiddle because the trench and the overlying layerare composed of the same material, resulting inessentially no thermal mismatch between them.The localized stresses in the Cu and the oxide linesnear the interfaces are expected to nearly canceleach other out. Therefore, curvature changes acrossthe lines from the homogenized composite layer,Djy, which are related to the volume-averagedstresses of the overall layer, hryyi, with contribu-tions from both Cu and oxide lines, show goodagreement with the theory even down to aroundh/w = 1 (Fig. 6).22 When the aspect ratio is muchhigher (h/w = 4), the area of the regions wherelocalized stresses are found is smaller, which mayexplain why FEM results approach the theoreticalprediction with increasing aspect ratio.

Effect of Passivation Materials

Table II compares each volume-averaged normalstress component in the Cu lines passivated witheither a SiO2 or a Si3N4 layer. In addition to anincrease in hrl

yyi; there is a large increase in hrlzziwhen

passivated with a Si3N4 top layer. In the high-aspect-ratio limit, isostress condition in the y-direction andisostrain condition in the z-direction are assumed.When passivated with a capping layer, isostress con-dition across the lines is not strictly valid because oflocalized stress near the interface between the linesand the passivation layer. Passivation, however, cau-ses a favorable effect on the isostrain condition in theout-of-plane direction. These effects in both the y- andz-directions become more pronounced when a stiffercapping layer is deposited.

Figure 10a shows a contour of rzz in a Cu line and anoxide trench (h/w = 2 and fl = 0.5) when passivated

with a SiO2 layer (hp/h = 1). Near the interfacesbetween under/overlying layers, an isostrain condi-tion in the z-direction does not work well since Cu islikely to have a higher strain than the oxide due to itsmuch higher thermal expansion coefficient relative toSiO2, even when the lines are passivated with cap-ping materials. The current theory assumes that thestrains in the z-direction should be the same, whichresults in very stiff confinement of the Cu lines in thedirection normal to the substrate because Cu tends toexpand more than oxide. This leads to an overesti-mation of the volume-averaged stresses of the Culines in the z-direction, hrl

zzi; compared to the FEMresults, as shown in Fig. 8c. Figure 10b shows acontour of rzz when passivated with Si3N4. Because ofits higher stiffness, it confines the Cu lines in thez-direction more than the SiO2 passivation layer.Therefore, it leads to an increase in hrl

zzi; and theFEM value becomes much closer to the theoreticalprediction.

Table II. Comparison of Theoretical Predictions ofVolume-Averaged Stresses to Finite Element

Results in Cu Lines on Single-Level StructuresWithout Passivation and with SiO2 or Si3N4

Passivation (w/p = fl = 0.5, h/w = 2, hp/h = 1, h = 1 lmand hs = 525 lm) During Cooling from 200�C toRoom Temperature (Differences from Theory

Shown in Parentheses)

hrxxl i

(MPa)hryy

l i(MPa)

hrzzl i

(MPa)hrh

l i(MPa)

Theory 377 146 180 234FEM (unpass.) 387 (3%) 161 (10%) 130 (-27%) 226 (-3%)FEM (SiO2) 383 (2%) 171 (17%) 157 (-13%) 237 (1%)FEM (Si3N4) 389 (3%) 179 (23%) 169 (-6%) 245 (5%)

Fig. 9. 2-D FEM contours of stress across lines, ryy, on single-level structures with SiO2 passivation layers (w/p = fl = 0.5, h = 1 lm, hp = 1 lmand hs = 525 lm) at different aspect ratios during heating from room temperature to 200�C: (a) h/w = 1, (b) h/w = 2, and (c) h/w = 4.


785

Effect of Thermoelastic Properties of Low-kDielectrics

Based on the analytical solution in this work,volume-averaged thermal stresses in copper inter-connect lines and dielectric lines can be computedfor lines that are sufficiently tall. Since most low-kdielectrics replacing oxides are polymer-basedmaterials, the relevant elastic constants are typi-cally low. Hydrostatic stresses in lines in a widerange of thermoelastic properties of low-k dielec-trics during cooling from 200�C to room tempera-ture are shown in Fig. 11. Assuming that thePoisson�s ratios of the dielectrics and the oxide arethe same, while they are insensitive to the coeffi-cients of thermal expansion (CTE) of the dielec-trics, the thermal stresses in the Cu lines shownin Fig. 11a scale with stiffness of dielectrics, whichis in agreement with previous finite elementresults.27 Stresses in the dielectrics (Fig. 11b),however, depend on the CTEs as well as the elasticmoduli of the dielectric materials. Correspondingstresses from some candidates for low-k dielec-trics28 are indicated to illustrate how the analyticalexpressions and such maps can be used for selec-tion of dielectric materials in integration with Cuinterconnects. Either of a high-modulus/low-CTEmaterial such as tetraethylorthosilane (TEOS) or alow-modulus/high-CTE material such as hydrogensilsesquioxane (HSQ) is compatible with the Culines, as far as the thermal stresses in the dielec-tric lines are concerned.*

Effect of Metal Density

Figure 12 shows thermal stresses in Cu lines ata fixed (and high) line aspect ratio (h/w = 4) withSiO2 passivation (hp/h = 1). Lines with high aspectratio are chosen because they represent the cur-rent industrial trend for line geometry in themodern IC industry. Fortuitously, in the high-aspect-ratio case, we can also minimize the effectsof localized stresses from under/overlying layers.The validity of the theory at high aspect ratios(tall lines) can then be evaluated explicitly overa wide range of metal density. hrl

xxi remainsnearly constant with metal density. As metaldensity increases, however, hrl

yyi increases andhrl

zzi decreases. As the line structure approachesa continuous Cu film (w/p = 1), an equibiaxialstress condition ðhrl

xxi ¼ hrlyyiÞ with plane stress

ðhrlzzi ¼ 0Þ is obtained. For tall lines, the current

model is in very good agreement with FEMresults, especially when the spacing between lines issmall. It does, however, show deviation from FEMvalues as the spacing becomes larger, since oxidelines no longer have high aspect ratios. The abilityto predict the effect of metal density at a fixedaspect ratio is one of the advantages of the cur-rent model over the Eshelby-type inclusion theory,which is unable to take into account the presence ofneighboring lines.15,16 The stress model for passiv-ated metal lines proposed by Wikstrom et al.,17

however, produces very similar values as the modelin the current work, over a wide range of metal den-sity. Although their model cannot predict curvatureevolution, the assumption that in-plane deformationin lines are totally controlled by substrate deforma-tion seems good as far as volume-averaged stressesare concerned.

*Compatibility issues of low-k dielectrics with Cu metalizationassociated with stress concentration in via structures will beaddressed in another paper.29

Fig. 10. 2-D FEM contours of stress normal to the substrate, rzz, on single-level structures with different passivation layers (w/p = fl = 0.5,h/w = 2, hp/h = 1, h = 1 lm and hs = 525 lm) during heating from room temperature to 200�C: (a) SiO2 passivation and (b) Si3N4 passivation.


Comparison with Experiments

Table III compares theoretical predictions ofnormal stress components in Cu lines with numer-ical simulation results in the current work andX-ray diffraction data in a previous report.8 Sincethe line aspect ratio of this sample is not exceed-ingly high (h/w = 0.7), it is difficult to comparedirectly with the theoretical prediction in the high-aspect-ratio limit. The theory�s underestimationof hrl

yyi and overestimation of hrlzzi compared to

FEM values have been discussed previously in thesections ‘‘Effect of Line and Passivation Geometryon Stress Evolution’’ and ‘‘Effect of PassivationMaterials.’’ Hydrostatic stress, however, which isusually taken to be a key parameter for electromi-gration simulation,5 is quite well predicted. Finiteelement results capture the trend of stress valuesfrom X-ray measurement in spite of the complicatedline geometry, especially mixed passivation layers of

SiO2 and Si3N4. The differences between FEM andexperimental results are due to inelastic deforma-tion, such as plastic yielding and/or creep in the linesand anisotropic material properties of Cu, whichare not considered in this analysis. It can also beattributed to the low spatial resolution of conven-tional X-rays (i.e., a spot size that is usually order ofmagnitude larger than the width of the lines).

Multilevel Structures

In order to capture three-dimensional (3-D) fea-tures of the multilevel structure, especially theperpendicular arrangement, a fully 3-D finite ele-ment analysis was performed. Results were ana-lyzed to determine whether the curvaturepredictions are valid for the multilevel structureand to find how thermal stresses vary from level tolevel. The effects of upper-level line arrangement onstress distribution in lower-level lines areaddressed. It is also discussed whether the analyt-ical prediction of thermal stresses based on the

Table III. Comparison of Theoretical Predictionsof Volume-Averaged Stresses in Cu Lines (with

0.1 lm Si3N4 and 0.8 lm SiO2) to Results of FiniteElement Analysis and X-ray Diffraction

Experiments (w/p = fl = 0.5, h/w = 0.7, hp/h = 1.3,h = 0.7 lm and hs = 525 lm) During Cooling from

390�C to Room Temperature

hrxxl i

(MPa)hryy

l i(MPa)

hrzzl i

(MPa)hrh

l i(MPa)

Theory 806 297 380 494FEM 727 431 273 477Experiment 539 459 222 407

Fig. 11. Analytical predictions of volume-averaged hydrostaticstresses as a function of thermoelastic properties of dielectricmaterials during cooling from 200�C to room temperature at fl = 0.5:(a) in Cu lines and (b) in dielectric lines.

Fig. 12. Volume-averaged stresses in Cu lines on single-levelstructures with a SiO2 passivation layer (h/w = 4, hp/h = 1, h = 1 lmand hs = 525 lm) as a function of metal density (w/p) during coolingfrom 200�C to room temperature.


787

theory in the limiting line geometry of the singlelayer can be generalized to provide accurate stressestimation in multilevel structures.

Calculation of Curvatures Using Superimposition

Table IV compares theoretical curvature predic-tions with 3-D FEM results in two different linearrangements following the dual Damascene pro-cess shown in Fig. 3. In this calculation, both lowerand upper levels have the same line structure(h/w = 1 and fl = 0.5). In the aligned structure(Table IVa) the superimposition of three layers, i.e.,two composite layers with lines running in the samedirection and one ILD layer between them, displaysvery good agreement of the theory (Eqs. 7 and 8)with the FEM values. In the perpendicular struc-ture (Table IVb) the same curvature values areexpected in the x- and y-directions according to thetheory due to the symmetry of the structures. Thisis an expectation which is confirmed by thenumerical simulation.

Because of the high diffusivity of Cu through SiO2

and Si, a diffusion barrier such as TaN is commonly

employed between the metal and the dielectriclines.19 Although this complicates the interconnectstructures, the effect of this additional layer caneasily be incorporated in the present model by lin-ear superimposition and may be negligible if thislayer is very thin, i.e., on the order of a few hundredAngstroms (e.g., less than 10% of the line width).

Effect of Line Arrangement on Stress Evolutionon Lower/Upper Levels

Figure 13 shows contours of hydrostatic stressesduring cooling from 200�C to room temperature(20�C) in a two-level Cu interconnect structurewithout a passivation layer on top of the upper level.This corresponds to the structure after polishing theexcess Cu layer on the upper level in the dualDamascene process. In this calculation, both lowerand upper levels have the same line structure(h/w = 1 and fl = 0.5). For the purpose of comparison, ahydrostatic stress contour of a single-level structurewith a passivation layer is shown in Fig. 13a. In thealigned structure (Fig. 13b) the stress values in theCu line on the upper level are much lower than

Table IV. Comparison of Theoretical Predictions of Curvature Changes of Trilayer Structures (Fig. 3) afterthe Dual Damascene Process to Finite Element Results (w/p = fl = 0.5, h/w = 1, hp = 1 lm, h = 1 lm, and

hs = 525 lm) During Cooling from 200�C to Room Temperature: (a) Aligned Structure and (b) PerpendicularStructure

jx (l/m) jy (l/m) jx (l/m) jy (l/m)

Theory 0.0394 0.0293 Theory 0.0343 0.0343FEM 0.0408 (+3.6%) 0.0297 (+1.2%) FEM 0.0355 (+3.5%) 0.0352 (+26%)

Fig. 13. Contours of hydrostatic stress, rhydro, during cooling from 200�C to room temperature (w/p = fl = 0.5, h/w = 1, hp/h = 1, h = 1 lm andhs = 525 lm): (a) single-level with SiO2 passivation using 2-D FEM, (b) two-level aligned structure, and (c) two-level perpendicular structure afterdual Damascene process using 3-D FEM.


those on the lower level. There are two main reasonsfor the above phenomenon. First, the lines on theupper level are farther from a stiff Si substrate,which relieves stresses, especially in the directionnormal to the substrate. In addition, there is nopassivation layer on the upper level of the structureafter polishing. Therefore, based on the earlierargument, an absence of the passivation layerwould reduce stress values in the Cu lines on theupper level. In the perpendicular structure(Fig. 13c) hydrostatic stresses in the lines on theupper level have almost the same values as those inthe aligned structure. Furthermore, hydrostaticstresses in the lines on the lower level are notheavily influenced by the presence of the upper leveland its arrangement.

Figure 14 shows the contours of von Mises stres-ses during cooling. Unlike hydrostatic stresses, vonMises stresses are higher in the line on the upperlevel, except at sharp corners. This is due to theabsence of the passivation layer on the upper levelbecause unpassivated Cu lines which are in thedielectric trench with a free top surface may havehigh deviatoric components unless the aspect ratiois very high. It can also be seen that the upper-levellines show the same results in both the aligned andperpendicular arrangements, and the lower-levellines are hardly affected by the presence and/orarrangement of the upper level. It should be noted,however, that the difference of von Mises stressdistribution in the ILD layer can be found in dif-ferent line arrangements. Typically, the stress val-ues are not exceedingly high, but in general, newinsulating materials with low dielectric constantshave very low yield strength,7 which can influence

other failure mechanisms, such as yielding in thedielectric.

For the structures in Figs. 13 and 14, volume-averaged values of each normal stress component,hydrostatic, and Mises stresses are summarized inTable V. Compared to a single-level structure, thestress components in the lower-level lines of two-level structures basically remain unchanged, andare independent of the arrangement of the upper-level lines. In the upper lines, however, hrl

yyi andhrl

zzi are smaller than those in the lower-levellines. To separate the effects of the relative posi-tions of the lines (whether lines lie on the lower orupper level) from effects of line geometry (whetherlines are passivated or not), SiO2 of the samethickness as the ILD layer was used for the pas-sivation layer on the upper level (Table VI). Theupper-level stress values are nearly identicalalong and across the lines (in-plane stress com-ponents) as the lower one, but give slightlysmaller values in the stress normal to the sub-strate (out-of-plane stress component). This is inagreement with a previous result from Shen.18

This can result from the fact that the upper-levellines are closer to the free surface, so the con-straint imposed by the surrounding oxide is not asstrong as that experienced by the lower-levellines. There is little difference between the alignedand perpendicular line arrangements. Therefore,it can be seen that interaction between levels inthe vertical direction is quite weak when thethickness of the interlevel dielectric (ILD) becomescomparable to that of a metal layer, which iscommonly found in practical multilevel structuresin the semiconductor industry.

Fig. 14. Contours of von Mises stress, rMises, during cooling from 200�C to room temperature (w/p = fl = 0.5, h/w = 1, hp/h = 1, h = 1 lm andhs = 525 lm): (a) single-level with SiO2 passivation using 2-D FEM, (b) two-level aligned structure, and (c) two-level perpendicular structure afterthe dual Damascene process using 3-D FEM.


789

Extension to Four-Level Structure

The finite element analysis of volume-averagedstresses can be extended to a four-level structurewith the same line geometry (h/w = 1, hp/h = 1 andfl = 0.5) as seen in Table VII. It is clear that thedifferences from level two to level four in the stresscomponents are negligible. Since Si is much stifferthan SiO2, the deformation of Cu on the first level ismore constrained than those on the other levels. Aslong as line aspect ratios are sufficiently low or high,this implies that the current theoretical predictionsof stress evolution based on the limiting line geom-etry in the single-level structure are applicable toany level of a multilevel structure, with a slightdifference in the out-of-plane stress in the first levelon the substrate.

CONCLUSIONS

On the basis of theoretical and numerical analy-ses carried out in the present work for the single-

and multilevel interconnect line structure, thefollowing conclusions are made:

� The interconnect structure comprised of metaland dielectric lines can be characterized by twolimiting cases. In the limit of very low aspectratios, each metal and dielectric line behaves likean individual thin film, and equibiaxial planestress conditions prevail in each line. When linesare sufficiently tall (high aspect ratio), the linestructure is homogenized into a single compositelayer with different effective thermoelastic prop-erties along and across the lines due to geomet-rical anisotropy. The 2-D finite element methodusing a generalized plane-strain model is in goodagreement with the theoretical predictions as theline geometry approaches these limiting cases andalso shows a gradual transition at intermediateline aspect ratios.

� When passivated with a capping layer, an iso-stress condition across the lines is not strictlyvalid because of localized stresses (mainly inter-facial shear stresses) near the interface betweenthe lines and the passivation layer. Passivation,however, has a favorable effect on the isostraincondition in the out-of-plane direction. Theseeffects of passivation become more pronouncedas the stiffness of the capping layer increases. Insufficiently tall lines, however, passivation doesnot make a difference to the stress states in thelines, and the analysis is extremely accurate.

� The current analytical stress model can predictthe effect of metal density. It can be shown thatthe interaction between lines on the same level,i.e., in the lateral direction, is so strong that itcannot be neglected.

Table V. Comparison of Volume-Averaged Stresses in Cu Lines on the Single-Metal-Level Structure withSiO2 Passivation and Two-Metal-Level Structures without Passivation (w/p = fl = 0.5, h/w = 1, hp = 1 lm,

h = 1 lm and hs = 525 lm) During Cooling from 200�C to Room Temperature

hrxxl i (MPa) hryy

l i (MPa) hrzzl i (MPa) hrh

l i (MPa) hrMisesl i (MPa)

FEM (single) 387 202 138 242 239FEM (double; align) Lower 386 200 139 242 237

Upper 375 170 81 209 280FEM (double; perp.) Lower 386 199 138 241 236

Upper 373 171 80 208 281

Table VI. Comparison of Volume-Averaged Stresses in Cu Lines on Single-Metal-Level Structure with SiO2

Passivation and Two-Metal-Level Structures with SiO2 Passivation (w/p = fl = 0.5, h/w = 1, hp = 1 lm,h = 1 lm, and hs = 525 lm) During Cooling from 200�C to Room Temperature

hrxxl i (MPa) hryy

l i (MPa) hrzzl i (MPa) hrh

l i (MPa) hrMisesl i (MPa)

FEM (single) 387 202 138 242 239FEM (double; align) Lower 386 199 139 242 237

Upper 383 198 131 237 236FEM (double; perp.) Lower 385 200 141 241 236

Upper 375 197 130 234 238

Table VII. Comparison of Volume-AveragedStresses in Cu Lines on Each Level from a

Four-Level Structure (w/p = fl = 0.5, h/w = 1,hp = 1 lm, h = 1 lm, and hs = 525 lm) DuringCooling from 200�C to Room Temperature

hrxxl i

(MPa)hryy

l i(MPa)

hrzzl i

(MPa)hrh

l i(MPa)

FEM (M1) 384 199 139 241FEM (M2) 381 196 132 236FEM (M3) 381 196 132 236FEM (M4) 381 196 131 235


� Comparison of the current model to availableexperimental data shows reasonable agreement,considering the complicated line geometry, inelas-tic deformation/anisotropic material properties inthe sample, etc.

� From 3-D FEM results of the two-level structure,there is little difference between the aligned andthe perpendicular line arrangements. Therefore,it can be seen that the interaction between levelsin the vertical direction is quite weak when thethickness of the interlevel dielectric (ILD) layerbecomes comparable to that of the metal layer.

� In a multilevel structure, the differences betweenthe stress components in the lines of the secondlevel to those of any other layer (all the way to thetop) are negligible. This implies that stress builtup in each level is not heavily influenced by thepresence of additional layers. This was found to betrue when both limiting cases (either flat or talllines) were approached.

ACKNOWLEDGEMENTS

We acknowledge support of the DefenseUniversity Research Initiative on Nano Technology(DURINT) on ‘‘Damage- and Failure-ResistantNanostructured and Interfacial Materials’’ which isfunded of MIT by the Office of Naval Researchunder Grant N00014-01-1-0808.

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