Numerical simulations and experimental

Numerical simulations and experimental measurements of stress relaxation by interface diffusion in a patterned copper interconnect structure N. Singh, A. F. Bower, D. Gan, S. Yoon, P. S. Ho et al. Citation: J. Appl. Phys. 97, 013539 (2005); doi: 10.1063/1.1829372 View online: http://dx.doi.org/10.1063/1.1829372 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v97/i1 Published by the American Institute of Physics.

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JOURNAL OF APPLIED PHYSICS 97, 013539 (2004)

Numerical simulations and experimental measurements of stress relaxation by interface diffusion in a patterned copper interconnect structure N. Singh and A. F. Bowera) Division of Engineering, Brown University, Providence, Rhode Island 02912

D. Gan, S. Yoon, and P. S. Ho Laboratory for Interconnect and Packaging, University of Texas, Austin, Texas 78712-1063

J. Leu and S. Shankar Logic Technology Development, Intel Corporation, 5200 N.E. Elam Young Parkway, Hillsboro, Oregon 97214

(Received 25 May 2004; accepted 14 October 2004; published online 16 December 2004) We describe a series of experiments and numerical simulations that were designed to determine the rate of stress-driven diffusion along interfaces in a damascene copper interconnect structure. Wafer curvature experiments were used to measure the rate of stress relaxation in an array of parallel damascene copper lines, which were encapsulated in a dielectric, and passivated with an overlayer of silicon nitride or silicon carbide. The stress relaxation was found to depend strongly on the choice of passivation. Three-dimensional finite element simulations were used to model the experiments, and showed that this behavior is caused by changes in the diffusivity of the interface between the copper lines and the passivation. By fitting the predicted stress relaxation rates to experimental measurements, we have identified the interfaces that contribute to stress relaxation in the structure, and have estimated values for their diffusion coefficients. © 2004 American Institute of Physics. [DOI: 10.1063/1.1829372] I. INTRODUCTION

Interconnect lines are thin metal wires that make electrical contact between devices on an integrated circuit. There have been several advances in interconnect technology in recent years: Aluminum interconnects have been replaced with copper; and dimensions have been reduced from 0.25␮m (1995) to 0.13␮m (2002). Current industry roadmaps seek to progressively reduce interconnect dimensions to 0.065␮m, and to replace the passivation with materials that have a lower dielectric constant. Each time such a change is made, the material systems must be tested to ensure that they can withstand mechanical, thermal, and electrical loading during service. Stress and electric currentdriven diffusion along interfaces between interconnect lines and the surrounding passivation is a particular concern, since voids and cracks tend to form at points of flux divergence in the line, causing open circuits. The current generation of copper interconnects are generally manufactured using the dual-damascene method, in which the various components of the structure are fabricated using a complex sequence of deposition, etching, electroplating, and chemical–mechanical polishing of successive layers of material.1 A typical resulting structure is illustrated schematically in Fig. 1. Dual-damascene copper interconnects are encapsulated on three sides by a thin layer of tantalum or tantalum nitride/tantalum bilayer, which acts as a diffusion barrier to copper. One face (above or below) of the interconnect is covered by an etch-stop layer of either silicon carbide or silicon nitride. Results from electromigration studies suga)

Electronic mail: [email protected]

0021-8979/2005/97(1)/013539/11/$22.50

gested that the interface between Cu and the nitride or carbide cap layer is the dominant path for electromigrationinduced mass transport in the structure.2 There is therefore great interest in devising ways to measure, and ultimately to control, diffusion along this interface. Very few techniques can quantitatively measure the diffusivity of interfaces in representative interconnect structures. Instead, diffusivities are determined indirectly, through tests that measure electromigration lifetimes: Structures with a low resistance to electromigration are assumed to have a fast diffusion path, and postmortem examination of failed test structures can reveal locations of flux divergences. Electromigration tests show that voids in dual-damascene copper usually form at the copper/etch-stop interface. In addition, Lane et al.3 and Hu et al.4 have shown that modifying this interface has a strong influence on electromigration lifetimes. A more direct test of interface diffusivity was devised recently by Gan et al.,5 who used wafer curvature experiments to measure the relaxation of stress in a patterned in-

FIG. 1. A schematic illustrating a cross section through a typical dualdamascene copper interconnect structure. The thickness of the various layers is not to scale. 97, 013539-1

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terconnect structure following cooling from 430 °C to 200 °C. Their samples contained parallel damascene lines encapsulated on three sides by Ta/TaN barrier layers, surrounded by an SiOF dielectric, and passivated on the top surface by a layer of either SiC or SiN. The results of these tests showed that changing the passivating layer has a significant influence on stress relaxation: The unpassivated lines relax quickly; those with a nitride passivation relax slowly, while the carbide passivated structures relax at an intermediate rate. The relative rates of stress relaxation correlate well with measured electromigration lifetimes of these structures, leading the authors to conclude that differences in stress relaxation rates were a consequence of changes in the diffusivity of the interconnect/passivation interface. These results suggest that stress relaxation measurements in patterned structures can be exploited as a rapid means of assessing interface diffusivity. The method has a limitation, however, in that it is difficult to determine quantitative values of interface diffusivity, which would be required in any quantitative model of interconnect reliability. The principal difficulty is that patterned line structures contain a complex three-dimensional distribution of stress, and it is impossible to relate measured stress relaxation to interfacial properties using any simple analytical model. To allow a more quantitative interpretation of these experiments, we have developed a three-dimensional finite element method to model stress relaxation by interface diffusion in elastic and inelastic solids, and have used this method to model stress relaxation in the test structures used by Gan et al.5 The results of our calculations show good agreement with the experimental trends. In particular, we find that the interface diffusivities in the test structures have a significant effect on the measured stress relaxation, suggesting that a systematic comparison of measured relaxation data with numerical simulations can yield accurate measurements of mass transport rates required in reliability models. The remainder of this article is organized as follows. In the next section, we give a brief review of stress relaxation mechanisms in interconnects as background to our study. In Sec. III, we briefly describe the experimental procedure for measuring stress relaxation. In Sec. IV, we outline the threedimensional finite element method used to simulate stressdriven interface diffusion. Section V describes the results of the experiments and numerical simulations, and Sec. VI contains conclusions. II. STRESS RELAXATION MECHANISMS IN INTERCONNECTS

Since the pioneering work of Korhonen and co-authors,6,7 the mechanisms of stress relaxation in interconnects have been studied extensively, both experimentally and theoretically. We emphasize that our goal in this article is not to develop and verify experimentally a general-purpose computer-aided design tool to predict stresses in interconnects. Rather, we intend to use a combination of experiment and theory to elucidate the mechanisms of diffusional stress relaxation in the current generation of copper interconnects. Nevertheless, a brief review of stress relaxation mechanisms is helpful background to our work.

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There are four possible inelastic mechanisms for relaxing thermal stresses in interconnects: (i) Diffusion through grain boundaries; (ii) diffusion along the interface between the line and the surrounding passivation; (iii) diffusion through the lattice; and (iv) conventional plastic flow due to dislocation motion. The dominant mechanism depends strongly on (a) temperature; and (b) the line dimensions. A detailed theoretical discussion of the transitions from one mechanism to another is given, e.g., in Ref. 8. Briefly, at modest stresses and high temperatures, diffusional flow is the dominant stress relaxation mechanism, regardless of linewidth. In the old generation of aluminum–copper interconnects, which had linewidths of 2 µm, grain boundaries provided the dominant diffusion path. The current generation of copper interconnects have linewidths of only 0.13 µm. These lines have a nearly perfect bamboo grain structure, with grain boundaries oriented perpendicular to the line direction. Diffusion along bamboo grain boundaries cannot relax stress, so in such structures the interfaces between lines and passivation are the dominant diffusion path. Plastic flow by dislocation motion also plays an important role in relaxing stress, particularly in blanket films, and in the old generation of 2 µm Al–Cu lines. Plastic flow tends to dominate at high stress levels, and lower temperatures, where diffusion is more effectively suppressed. Detailed discussions of stress relaxation by plastic flow can be found, e.g., in Park and Jeon,9 Senez et al.,10 and references therein. Plastic flow is highly linewidth dependent due to the constraining effects of the lateral boundaries of the line on the nucleation and propagation of dislocations.11–13 There is strong experimental evidence to suggest that the current generation of copper interconnects (with 0.13 µm linewidth) remain elastic under sufficiently rapid thermal cycling. For example, Rhee et al.14 have reported x-ray measurements of the stresses in 0.13 µm copper lines, under thermal cycling between 50 °C and 400 °C, with a cooling rate of 2 °C per minute. In this test, there was not sufficient time for diffusion to significantly relax stresses in the line. In addition, plastic flow was suppressed, due to the fine linewidth. Consequently, the experimental data show that the lines remain perfectly elastic over the full range of temperatures. The line structures used in our study are identical to those tested by Rhee et al.14 In contrast to their experiments, however, our focus is to study the stress relaxation that occurs when the lines are cooled from 430 °C to 200 °C. This induces relatively modest stresses in the lines, which are well below the level required to induce plastic flow by dislocation motion. Instead, the stresses relax slowly by diffusion along grain boundaries and interfaces in the solid. By measuring the changes in stress during relaxation, and comparing the results to calculations, we shall attempt to identify the dominant diffusion paths in the structure, and to estimate values for diffusion coefficients for the critical interfaces. Stress-driven diffusion is, of course, a mechanism of high-temperature creep (Coble creep, if diffusion occurs primarily along grain boundaries; or Nabarro–Herring creep if relaxation is by lattice diffusion). It is common to model creep using phenomenological constitutive equations: The usual models of Coble or Nabarro–Herring creep use a linear


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FIG. 3. Unit cell used in finite element computations.

FIG. 2. (a) A schematic diagram illustrating the patterned line structures used in experimental measurements. (b) A cross section through the experimental test structure.

relationship between strain rate and stress. This approach is not used in the simulations reported here. Instead, we model diffusion along grain boundaries explicitly, by solving the coupled equations governing elastic deformation within each grain and stress-driven diffusion along grain boundaries. The grains themselves are modeled as elastic solids, and a linear diffusion equation is used to model mass transport along grain boundaries. The relationship between average strain rate and average stress in the structure is an outcome of the calculations, rather than an a priori assumption. Our numerical simulations will be described in more detail in Sec. IV. III. EXPERIMENTAL MEASUREMENTS

Gan et al.5 have conducted a series of experiments to determine the rate of stress relaxation by interface diffusion in representative three-dimensional interconnect structures. Their experiments have been described in detail elsewhere, and so will only be briefly summarized here. A schematic diagram illustrating the specimen is sketched in Fig. 2. The structure consists of a parallel array of damascene copper lines, with width 0.2 µm, height 0.6 µm, and spacing 0.2 µm. The grain size and texture of the lines was not determined. It is likely that the lines have a “bamboo” grain structure, with grain size approximately equal to the linewidth, and with most grain boundaries oriented perpendicular to the line direction, as indicated in the sketch. There have been several studies of texture in damascene copper lines and films.15,16 In most cases texture is predominantly (111), although line structures with a significant (100) texture have also been suggested.16 The lines are surrounded on three sides by a Ta/TaN barrier layer, with a

thickness of 5 nm at the sidewalls and 30 nm at the base. They are encapsulated in a SiOF dielectric, which rests on a 0.5 µm thick film of SiO2. The entire structure is on a 250 µm thick substrate of Si, and is passivated on the top surface by a 180 nm thick film of either SiC or SiN. For comparison, some tests were also conducted with unpassivated structures. The specimens were subjected to a thermal cycle consisting of heating to 430 °C at a ramp rate of 4 °C per minute, followed by cooling at 2 °C per minute to 200 °C. Once the specimens reached the final temperature, an optical technique was used to measure the curvature of the substrate at 1 min intervals for a period of 2700 min. Curvatures ␬x and ␬y were measured parallel and transverse to the line direction, respectively. The average stresses acting parallel and perpendicular to the lines was computed using the generalized Stoney equation as

␴xx =

␴yy =

Ests2 6共1 − ␯s2兲t f Ests2 6共1 − ␯s2兲t f

共␬x + ␯s␬y兲,

共␬y + ␯s␬x兲.

共1兲

where Es, ␯s, t f , and ts denote Young’s modulus, Poisson’s ratio, film thickness, and thickness of the Si substrate, respectively. The results of these tests showed a significant influence of the passivating layer on stress relaxation: The unpassivated lines relaxed quickly; those with a nitride passivation relaxed slowly, while the carbide passivation relaxed at an intermediate rate. The relative rates of stress relaxation correlate well with measured electromigration lifetimes of these structures, suggesting that differences in stress relaxation rates were a consequence of changes in the diffusivity of the interconnect/passivation interface. Our goal in this article is to model stress relaxation in these structures, to elucidate the role of interface diffusion in stress relaxation, and in the hope of obtaining quantitative estimates of diffusivities for the various interfaces in the structure. Our numerical computations are described in more detail in the next section. IV. THEORY AND NUMERICAL PROCEDURE

Figure 3 shows an idealized model of the threedimensional interconnect structure described in the preceding section. The actual test structures are perfectly periodic in the direction transverse to the lines, and approximately periodic (for a bamboo grain structure) along the line. In our


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TABLE I. Dimensions used in modeling. Grain size, 2L = 0.7␮m. Material/layer

Thickness (µm)

Copper ILD (SiOF) Barrier layer (Ta) Base layer 共SiO2兲 Passivation layer (SiN or SiC)

0.675 0.680 0.005 0.5 0.18

TABLE II. Material properties used in numerical simulations.

Width, 2w 共␮m兲 0.2 0.2 ¯ ¯ ¯

Material

simulations, we assume perfect periodicity in both directions, in which case it is necessary to model only one-quarter of a representative grain, with appropriate symmetry boundary conditions on the sides of the simulation cell. The actual region modeled is illustrated in Fig. 3, and dimensions of the various parts of the structure are listed in Table I. The surface of the SiC or SiN cap layer is assumed to be free of tractions, and the base of the Si region is constrained to remain fixed. The copper interconnect, its Ta barrier layer, the dielectric, the substrate, and cap layer are all idealized as elastic single crystals. The materials are separated by planar sharp interfaces. In addition, a grain boundary, with orientation perpendicular to the line direction, lies at the symmetry plane at x = L. In our model, we assume that atoms can diffuse along the Cu/Cu grain boundary, the interfaces between Cu and the surrounding Ta barrier layers, and the interface between the Cu and cap layer. The remaining interfaces do not allow mass transport. In the following, we let V denote the volume occupied by the solid, and let S denote the collection of interfaces and grain boundaries which allow mass diffusion. Introduce an arbitrary coordinate system 共␰ , ␩兲 on S, and at each point define an orthonormal basis 兵n , t共1兲 , t共2兲其 where n denotes the normal to S and t共␣兲, where ␣ = 1, 2, denotes two tangent vectors on S. We suppose that the structure is free of stress at some initial reference temperature T0. The solid is then heated or cooled to a temperature T1, inducing a displacement field u共x兲 and stress field ␴ in the solid. The displacement field is continuous within the solid, but may have jump discontinuities across the interfaces S. Let u±共x兲 = lim⑀→0 u共x ± ⑀n兲 denote the limiting values of displacement on each side of S, and let ⌬n = 共u+ − u−兲 · n, ⌬t␣ = 共u+ − u−兲 · t␣ denote the normal and tangential displacement jumps across S. In addition, let ␴n = n · ␴ · n, ␴t␣ = n · ␴ · t共␣兲 denote the normal and tangential tractions acting on S. We idealize the behavior of the solid using a standard linear elastic constitutive law. The displacement field is determined from the Navier equation of elasticity Cijkl关ukl − ␣kl共T1 − T0兲兴,j = 0,

Young’s modulus (GPa)

共2兲

where the comma denotes differentiation, in the usual manner, while Cijkl and ␣kl denote the elastic modulus tensor and thermal expansion coefficient, respectively. In actual computations, we idealize the copper lines as cubic crystals, with anisotropic elastic constants listed in Table II. The remaining materials are assumed to be isotropic: Values for Young’s modulus and Poisson’s ratios for the various parts of the structure are listed in Table II.

SiOF (ILD) Ta SiN SiC Si Cu a

Coefficient of thermal expansion

Poisson’s ratio 共␯兲

−6

71.7 0.94⫻ 10 0.16 0.343 125 17.7⫻ 10−6 300 3.3⫻ 10−6 0.24 440 4.5⫻ 10−6 0.17 162.6 2.6⫻ 10−6 0.278 Anisotropic with constants C11 = 168.0 GPa, C12 = 121.4 GPa, C44 = 75.5 GPa, and ␯ = 0.344.a

See Ref. 17.

Our principal goal in this article is to model the stress relaxation that occurs due to diffusion along the interfaces. We model mass transport along the interfaces using a conventional linear diffusion law, so that

j=−

DI␦I exp共− QI/kT兲 ⵜ s␮ kT

共3兲

denotes the volumetric flux of atoms along an interface. In Eq. (3), DI exp共−QI / kT兲 denotes the coefficient of interface diffusion; ␦I denotes the thickness of the diffusion layer, k denotes the Boltzmann constant, ⵜs denotes the surface gradient operator, and ␮ denotes the chemical potential of mobile atoms in the interface. Since the interfaces in our model are all planar, the appropriate chemical potential is −⍀␴n, where ⍀ is the atomic volume. A flux divergence causes atoms to attach or detach from the two crystals adjacent to the boundary, and so causes a velocity discontinuity w˙diff n = − ⵜs · j

共4兲

across the interface. The chemical potential, and hence the normal stress ␴n, must be continuous at triple junctions between interfaces. In addition to diffusion, our computations may also account for viscous sliding on the interfaces. Sliding is modeled using a linear viscous law w˙tsliding = ␴ t␣/ ␩ , ␣

共5兲

where ␩ is the viscosity of the interface. In all computations reported here, we use a very large value for ␩ and so effectively suppress interface sliding. Finally, we have found that numerical computations are simplified if a small linear elastic compliance is included in the response of the interface. Denoting the normal and tangential interface stiffness by kn and kt, respectively, we find that the total normal and tangential velocity discontinuity are related to the normal and tangential traction acting on the interface by

˙n= w

1 ⍀DI␦n exp共− QI/kT兲 ⵜ s · ⵜ s␴ n , ␴˙ n − kn kT


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˙ t␣ = w

1 1 ␴˙ t␣ + ␴t␣ . kt ␩

共6兲

Equations (2) and (6), together with the continuity conditions at triple junctions, and appropriate boundary conditions on the displacement field and volumetric flux at symmetry boundaries, completely determine the displacement and stress fields in the solid. The displacement and stress fields can easily be computed using the finite element method, suitably extended to account for interface diffusion. Our development follows the two-dimensional finite element method introduced in Bower and Craft,18 but some modifications are necessary to apply the method in three dimensions. We use a mixed finite element method, which computes both the displacement field u and the variation of normal and tangential stress ␴n, ␴t␣ over the interface S. We suppose that the displacement and stress fields are known at some time t, and wish to compute the change in displacement ⌬u and change in stress ⌬␴n, ⌬␴t␣ during a subsequent time interval ⌬t. The equilibrium equations (2) are expressed using the equivalent principle of virtual work

冕

Cijkl关⌬uk’l − ␣ij⌬T兴␦ui’jdV +

V

+

+

⌬␴n共␦u+i − ␦u−i 兲nidA

S

冕

⌬wn␦␴ndA +

冕

⌬wt␣␦␴t␣dA = 0.

S

冕

冕

⌬␴t␣共␦u+i − ␦u−i 兲t␣i dA

S

共7兲

S

Substituting for ⌬wn and ⌬wt␣ from Eq. (6), integrating by parts, observing that the net flux at triple junctions sum to zero, and that the flux into interfaces at symmetry boundaries vanishes, we obtain

冕

Cijkl关⌬uk,l − ␣ij⌬T兴␦ui,jdV +

V

+

冕 S

+

冕 S

1 ⌬␴n␦␴ndA kn

冕

⌬t

冕

⌬␴t␣共␦u+i − ␦u−i 兲t␣i dA +

冕

⌬t

S

+

S

冕 S

␩

␴t␣␦␴t␣dA = 0.

␴n = qn + ␬n共u+i − u−i 兲ni ␴t␣ = qt␣ + ␬t共u+i − u−i 兲ti␣ ,

共9兲

where ␬n and ␬t are two small stiffnesslike numerical parameters, and qn and qt␣ are interfacial degrees of freedom. Substituting into Eq. (8) yields a variational equation that can be solved for the variation of qn and qt␣ along interfaces, together with the displacement field ui共x j兲 within the solid, using the finite element method. In our implementation of the finite element method, the displacement field within the solid is interpolated between nodal values using standard linear hexahedral elements. The variation of qn, qt␣ over the interfaces S is approximated using planar 12 noded quadrilateral elements. On each of these elements, four nodes have displacement degrees of freedom, and are attached also to elements within the solid above the interface; four nodes have displacement degrees of freedom, and are attached to elements within the solid below the interface, and four nodes have qn, qt␣ as degrees of freedom. The variation of qn, qt␣ within each interface element is computed using a linear interpolation between these nodal values. Substituting in Eq. (8) and evaluating the integrals yields a system of linear equations for the displacement increments at each node and for ⌬qn, ⌬qt␣. These are augmented by appropriate constraints on the displacement field at symmetry boundaries, while continuity of normal stress at triple junctions is enforced using Lagrange multipliers. The equations may be solved repeatedly to calculate the variation of displacement and stress in the solid as a function of time. Naturally, since the stiffness matrix is constant (for a constant time increment) it needs only to be factored once. We have used this numerical technique to model stress relaxation in the interconnect test structures. By comparing the results to experimental measurements, we hope to obtain quantitative estimates of mass transport rates along the various interfaces in the structure. We proceed to describe the results of these computations. V. RESULTS AND DISCUSSION

⍀DI␦I exp共− QI/kT兲 ⵜs␴n · ⵜs␦␴ndA kT

S

+

⌬␴n共␦u+i − ␦u−i 兲nidA

rigid-body modes in parts of the mesh that are constrained only by Lagrange multiplierlike elements. These difficulties may be avoided by changing variables to define

1 ⌬␴t␣␦␴t␣dA kt 共8兲

Equation (8) was used by Bower and Craft18 as the basis for two-dimensional finite element computations of stress driven interface diffusion. In three dimensions, however, we have found that Eq. (8) cannot be solved easily for small values of ⌬t using standard finite element equation solvers, due to

The curvature measurements outlined in Sec. III were used to determine the average stress in the interconnect structures illustrated in Fig. 2, following a cycle of thermal loading. Specifically, the structures heated to 430 °C at the rate of 4 °C per minute, then cooled at 2 °C per minute to a lower temperature (either 240 °C, 200 °C, or 160 °C). The cooling induces thermal residual stress in the lines. The state of stress is complex: It is primarily tensile in the copper, and compressive in the surrounding dielectric. Subsequently, the stress relaxes slowly. The stress relaxation is monitored experimentally by measuring the change in curvature of the substrate as a function of time after cooling. The curvature is related to the change in stress in the interconnect, dielectric and cap layer by the Stoney equation. Naturally, this procedure does not provide a complete picture of the stress distri-


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FIG. 4. Measured stress relaxation in an array of damascene copper lines. Results are shown for films passivated with SiN and SiC, as well as for unpassivated lines; ␴yy and ␴xx are the components of stress parallel and transverse to the lines, respectively.

bution in the lines, partly because wafer curvature measurements can only detect the average stresses in the patterned structure, and partly because wafer curvature measurements can only detect changes in stress, and do not provide absolute values. If needed, absolute values of average stress could be inferred from the data reported in Rhee et al.,11, but in the following discussion we shall focus only on the changes in average stress during relaxation, which can be determined accurately using wafer curvature tests. Figure 4 shows a typical experimental result. It shows the results of three separate experiments: In the first test, the structure was passivated with a SiN cap layer; in the second test, it was passivated with a SiC layer, and in the third, the structure was unpassivated. The figure shows the average stress parallel to the line direction (denoted by ␴xx) and transverse to the line direction 共␴yy兲 for each experiment. Several features of the experimental result are worth noting. First, the experiment shows that changing the passivation has a strong influence on stress relaxation. The unpassivated lines relax fastest; those passivated with carbide relax at an intermediate rate, while SiN passivated structures relax only slowly. Second, for comparison with simulations, we note that the longitudinal 共␴xx兲 and transverse 共␴yy兲 stresses relax at comparable rates, but in each case the longitudinal stress relaxes somewhat faster than the transverse stress. The relative rates of stress relaxation in the test structures correlate well with their electromigration lifetimes, suggesting that stress relaxation tests can at least provide a way to rank the relative electromigration resistance of various cap layers. The tests do not, however, provide a direct measure of the diffusivity of the interfaces in the solid. In fact, it cannot even be concluded with certainty that stress relaxation occurs by interface diffusion: There are several other possible mechanisms of stress relaxation, including plasticity within the copper, or interface sliding. Our hope is that numerical simulations will help to resolve some of these issues. Figure 5 shows the results of a representative numerical simulation. In this computation, the lines were idealized with the geometry shown in Fig. 3, and with material properties listed in Table II. The structure was assumed to be stress free at 430 °C, cooled rapidly to 200 °C and then held at fixed

FIG. 5. Predicted stress relaxation in the idealized interconnect structure as a function of time. (a) Ideal (100) texture and (b) ideal (111) texture.

temperature. The diffusivity values listed in Table III were assigned to the various interfaces in the structure: These values were selected to give a good fit to the experimental measurements. Figure 5 shows the predicted average stress in the patterned film as a function of time after reaching 200 °C. Results are shown for both SiN and SiC passivations (approximated in the simulations by assigning appropriate values of modulus to the cap layer, and by modifying the diffusivity of the copper/cap layer interface), and also for an unpassivated structure. In the latter simulation, the cap layer was removed entirely, and surface diffusion was assumed to be sufficiently rapid to fully relax the stress at both the triple junctions connecting the surface and Cu/Ta interface, and the junction between the surface and grain boundary transverse to the line. Since the line texture is unknown, we show results for both (100) and (111) orientations for the Cu grain. TABLE III. Estimated values for interface diffusivities.

Interface

Diffusivity value at 200 °C 关D␦ exp共−Q / kT兲兴共m3 / sec兲

Cu/Cu Cu/Ta Cu/SiN Cu/SiC

6.72⫻ 10−27 6.72⫻ 10−30 6.72⫻ 10−29 6.72⫻ 10−28


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FIG. 7. Diffusion paths that cause stress relaxation.

FIG. 6. The influence of changes in elastic modulus of the passivation. The Cu/cap interface has a diffusivity of 共D␦兲exp共−Q / kT兲 = 6.72⫻ 10−28 m3s−1 for both simulations.

The predicted stress relaxation is in good qualitative agreement with the experimental results shown in Fig. 4. In particular, in each case the stress component parallel to the line direction 共␴xx兲 relaxes faster than the component transverse to the line 共␴yy兲. The simulations predict stress relaxation rates and values very similar to the experimental measurements. In addition, with the interface and grain boundary diffusion coefficients listed in Table III, the simulations predict the correct relative rates of stress relaxation for the SiN, SiC, and unpassivated lines. The stress relaxation rates are evidently highly sensitive to the grain orientation: For the (100) orientation, the two stress components relax at similar rates, while for the (111) orientation, the stress component parallel to the line direction relaxes significantly faster than the component transverse to the lines. The (100) orientation gives the best fit to the experiments. It is likely that the experimental structures contain a mixture of (100) and (111) oriented grains, which would give behavior between the two extremes shown in Fig. 5. In principle, one could average the two extremes to provide a better fit to the experiments, but since the exact line texture is unknown we have not attempted to do so. Since our model contains a large number of adjustable parameters, it is of interest to explore the sensitivity of our predictions to these parameters. The influence of the diffusion coefficients for the various interfaces in the structure is of particular interest. We begin by demonstrating that the difference observed experimentally in stress relaxation in the SiC and SiN passivated structures is not caused by the difference in elastic modulus of the passivation. To this end, Fig. 6 shows the results of two simulations: In the first computation, the modulus of the passivation was set to 300 GPa (representative of SiN); in the second, it was set to 440 GPa (representative of SiC). The diffusion coefficient for the interface between the copper and cap layer was set to 6.72 ⫻ 10−28 m3s−1 for both simulations. There is clearly no change in stress relaxation rate in the two cases, showing that stress relaxation is insensitive to the stiffness of the cap layer. This result suggests that interface diffusion is the dominant stress relaxation mechanism in the test structures.

Of the several possible stress relaxation mechanisms discussed in Sec. II, only stress relaxation by diffusion through the interface between the copper and its cap is likely to be affected significantly by changing the passivation. We now turn to investigate the influence of the various diffusion coefficients on stress relaxation in the test structures. To interpret the results, it is helpful to understand the role of each interface in relaxing the stress. Figure 7 shows a sketch of one grain in the structure, and illustrates the paths for the mass transport. When the structure is first cooled from the annealing temperature to the test temperature, the grain is subjected to a high tensile stress both parallel and transverse to the line direction, but the vertical stress is much smaller, because the passivation is relatively thin, and has a stress free surface. Consequently, there is a strong driving force for atoms to migrate from the passivation/copper interface into either the copper/copper grain boundary (relaxing primarily the stress component parallel to the line) or into the copper/tantalum interface (relaxing the transverse stress). To relax the stress, atoms must migrate along both the cap layer, into either the grain boundary or the Cu/Ta interface. At least two interfaces must permit diffusion for this to be possible. Furthermore, the diffusion coefficient of the copper/cap interface will influence the rate of stress relaxation only if diffusion along this interface is the rate limiting process. This is the case only if the grain boundary, or the copper/tantalum interface has a higher diffusivity than that of the copper/cap layer. Evidence to support these observations is presented in Fig. 8, which shows the effects of changing the diffusivity of the copper/copper grain boundary. In each case, we show predicted stress relaxation as a function of time for three simulations: One with parameters (diffusion coefficient and modulus) intended to represent a SiC cap layer; a second intended to represent an SiN cap layer, and a third intended to represent an unpassivated structure. Figure 8(a) shows results with low value for the copper grain-boundary diffusion coefficient 共DCu/Cu␦Cu/Cu兲exp共−QCu/Cu / kT兲 = 6.72 ⫻ 10−29 m3 / s兲, while Fig. 8(b) shows results with a fast grain-boundary diffusivity 共DCu/Cu␦Cu/Cu兲exp共−QCu/Cu / kT兲 = 6.72⫻ 10−26 m3 / s. For simplicity, results are shown for isotropic grains, with Young’s modulus 128.84 of GPa and Poisson’s ratio of 0.344. Two features of Fig. 8 are of particular interest. First, note that in Fig. 8(a), there is no difference in the stress relaxation rate between SiN and SiC passivated lines. This is because diffusion along the Cu/Cu grain boundary is slower than along the Cu/cap interface, and consequently grain


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FIG. 8. The influence of copper grain-boundary diffusivity on the stress relaxation in patterned lines. Results are shown for isotropic grains with Young’s modulus E = 128.84 GPa and Poisson’s ratio ␯ = 0.344 (a) 共DCu/Cu␦Cu/Cu兲exp共−QCu/Cu / kT兲 = 6.72⫻ 10−29m3 / s. and (b) 共DCu/Cu␦Cu/Cu兲exp共−QCu/Cu / kT兲 = 6.72⫻ 10−26m3 / s.

boundary diffusion, rather than interface diffusion, is the rate-controlling mechanism. Since our experiments showed that changing the cap layer has a strong influence on stress relaxation, we conclude that diffusion must be slower along the Cu/cap interface than along the Cu/Cu grain boundary. Second, note Fig. 8 shows that the stress relaxation rates for the unpassivated lines are highly sensitive to the value of the grain-boundary diffusivity. In Fig. 8(a), stress relaxation is much slower than in our experiments, while in Fig. 8(b) it is much faster. We find that the copper grain-boundary diffusivity must be of order 共DCu/Cu␦Cu/Cu兲exp共−QCu/Cu / kT兲 = 6.72 ⫻ 10−27 m3 / s to give the best fit to our experiments. Diffusion coefficients for the Cu/SiC and Cu/SiN interfaces must be at least a factor of 10 smaller than the grain-boundary diffusivity to affect the stress relaxation rate. The best qualitative fit to experiment is obtained with 共DCu/SiC␦Cu/SiC兲 ⫻exp共−QCu/SiC / kT兲 = 6.72⫻ 10−28 m3 / s and 共DCu/SiN␦Cu/SiN兲 ⫻exp共−QCu/SiN / kT兲 = 6.72⫻ 10−29 m3 / s. We have also studied the influence of the diffusivity of the copper/Ta interface: representative results are shown in Fig. 9. Copper adheres strongly to Ta, so diffusion along this interface is likely to be slow. We find that our simulations substantially overestimate the rate of stress relaxation with values for the copper/tantalum interface diffusivity exceed-

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FIG. 9. Influence of Cu/Ta interface diffusion on stress relaxation. Results are for isotropic copper grains, other parameter values are listed in Tables I–III. (a) 共DCu/Ta␦Cu/Ta兲exp共−QCu/Ta / kT兲 = 6.72⫻ 10−29m3 / s and (b) 共DCu/Ta␦Cu/Ta兲exp共−QCu/Ta / kT兲 = 6.72⫻ 10−30m3 / s.

ing 共DCu/Ta␦Cu/Ta兲exp共−QCu/Ta / kT兲 = 6.72⫻ 10−29 m3 / s. Using 共DCu/Ta␦Cu/Ta兲exp共−QCu/Ta / kT兲 = 6.72⫻ 10−30 m3 / s appears to give a good fit to experimental data, but the diffusivity could be substantially lower, since stress relaxation is not influenced greatly if the copper/tantalum diffusivity is reduced to 共DCu/Ta␦Cu/Ta兲exp共−QCu/Ta / kT兲 = 6.72 ⫻ 10−31 m3 / s. The predicted stress relaxation rates are also sensitive to the grain size of the copper lines, but the effect is relatively minor, as illustrated in Fig. 10. Reducing the grain size increases the rate of stress relaxation, and also increases the total longitudinal stress relaxed. We would expect to see a substantial increase in the rate of stress relaxation if the grain size were smaller than the width of the interconnect, however. Under these conditions, the line would no longer have a bamboo grain structure, and some grain boundaries would be oriented parallel to the line direction. These grain boundaries provide a path for atoms to diffuse quickly along and across the lines, and the Cu/SiC or Cu/SiN interface would no longer be rate limiting. The stress relaxation in these structures would be comparable to that in the unpassivated lines. There is great interest in replacing the dielectric used in current interconnect structures with materials that have a lower dielectric constant. Several candidate materials have been identified, including nanoporous silicon and various polymers. Most candidate low-k dielectric materials have


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FIG. 12. Predicted stress relaxation in an interconnect structure with dielectric modulus 4.7GPa: (a) Ideal (100) texture and (b) ideal (111) texture. FIG. 10. Predicted stress relaxation in lines with various grain sizes: (a) 0.5␮m and (b) 0.9␮m.

significantly lower elastic moduli than conventional passivations. The dielectric modulus is expected to have a strong influence on the thermal and electromigration-induced stress in interconnect structures. We have therefore conducted some preliminary experimental measurements and numerical simulations to investigate the influence of the dielectric modulus on stress relaxation in our test structures. Figure 11 shows an experimental result for a carbondoped oxide (CDO) dielectric, with an estimated modulus of 4.47 GPa. The general trends are similar to those shown for

FIG. 11. The stress relaxation measured in interconnect structures with CDO dielectric.

the SiOF dielectric in Fig. 4: In particular, the rates of stress relaxation are comparable, and we observe a similar influence of the passivating film. There are some striking differences, however: For example, with the more compliant dielectric, the longitudinal stress relaxes much faster than the lateral stress. In addition, the total stress relaxed is substantially lower in the structure with CDO dielectric. Figure 12 shows numerical simulations with a dielectric modulus of 4.7 GPa, and with interface and grain-boundary diffusivities listed in Table III. The qualitative trends predicted by the numerical simulations agree with experimental observations. As before, simulations with a (100) line texture appear to fit experimental results better than those with a (111) texture; the best fit would be obtained with a mixture of (100) and (111) grains. We do not see a good quantitative match between theory and experiment for the compliant dielectric, however. In particular, simulations predict a very low lateral stress, and hence a much smaller change in lateral stress during relaxation, than is observed in experiments. There are several possible reasons for the discrepancy: For example, in our simulations, we have assumed that the diffusion coefficients of all interfaces are identical in both SiOF and CDO structures, in addition, in our simulations, we assume that the structure is fully relaxed after the thermal anneal at 400 °C, but relaxation may take significantly longer in the CDO structures. It is instructive to compare our estimates for diffusion coefficients with available data. There have been a number of


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measurements of grain-boundary diffusion in polycrystalline copper, which can be compared with our estimates for copper grain-boundary diffusivity. For example, Gupta et al.19 reported an activation energy of QCu/Cu 0.95 eV and preexponential DCu/Cu␦Cu/Cu 2.9⫻ 10−15 m3 / s in Cu–Sn alloys. This corresponds to 共DCu/Cu␦Cu/Cu兲exp共−QCu/Cu / kT兲 = 2.2 ⫻ 10−25 m3 / s at 200 °C. Surholt and Hertzig20 reported selfdiffusion in polycrystals with various degrees of purity, obtaining activation energies in the range 0.75⬍ QCu/Cu ⬍ 0.87, with corresponding pre-exponentials 3.89⫻ 10−16 m3 / s ⬍ DCu/Cu␦Cu/Cu ⬍ 1.16⫻ 10−15. These values give diffusion coefficients ranging from 5.1⫻ 10−25 to 3.8⫻ 10−24 at 200 °C. Finally, in a recent study, Gan et al.21 obtained DCu/Cu␦Cu/Cuexp共−QCu/Cu / kT兲 = 5.5⫻ 10−26 m3 / s at 200 °C from stress relaxation measurements in unpassivated electroplated blanket films. The substantial variations between measured values of diffusion coefficient are partly due to differences in processing and operating conditions between experiments. In addition, other than the work of Gan et al.,21 most prior measurements of diffusion have been made at high temperatures (between 450 and 750 °C), and must be extrapolated to estimate values at 200 °C. Consequently, while there is some evidence that grain-boundary diffusion is slower in the damascene patterned lines tested here than in bulk specimens, our value of DCu/Cu␦Cu/Cuexp共−QCu/Cu / kT兲 = 6.72⫻ 10−27 m3 / s is within experimental uncertainty. We are aware of only one study of diffusion along interfaces that are comparable to those present in our structure: Gan et al.21 used measurements of stress relaxation in passivated blanket films to estimate the diffusivity of Cu/SiN interfaces, obtaining 共DCu/Ta␦Cu/Ta兲exp共−QCu/Ta / kT兲 = 1.1⫻ 10−27 m3 / s. Our best fit value is roughly a factor of 20 lower 共6.72 ⫻ 10−29 m3 / s兲 which again suggests that diffusion is slower in damascene patterned lines than in blanket films. We conclude with two observations. First, since we have been able to obtain estimates for the diffusion coefficients for the interfaces in the patterned interconnect structures at 200 °C, it is natural to speculate that it may be possible to determine activation energies for diffusion by measuring stress relaxation at a range of temperatures. We have not so far been able to measure stress relaxation over a sufficiently large range of temperatures to allow this, however, so it is left for future study. Second, our conclusion that the Cu/ passivation interface has a lower diffusivity than copper grain boundaries may appear counterintuitive, since electromigration tests suggest that the Cu/passivation interface is the dominant path for mass transport.3 These apparently contradictory observations are easily reconciled by noting that the electric field in an interconnect acts parallel to the Cu/ passivation interface, but perpendicular to the bamboo grain boundaries. Consequently, there is a large driving force for electromigration along the copper/passivation interface, whereas the driving force for diffusion along the grain boundaries is negligible.

they can withstand electrical, thermal, and mechanical loading during service. It is of particular interest to determine the rate of stress or electromigration-induced diffusion along interfaces between the interconnect and surrounding dielectric or etch-stop layer. To this end, we have used a combination of experimental measurements and numerical simulations to probe the rate of stress-induced diffusion along interfaces in a representative damascene interconnect structure. Wafer curvature measurements were used to determine the rate of stress relaxation in an array of copper lines following a cycle of thermal stress. Three types of structure were tested: Unpassivated lines; lines passivated with SiC, and lines passivated with SiN. (i)

(ii)

(iii)

(iv)

(v)

Experiments show a strong influence of the choice of passivation on the rate of stress relaxation in the lines. Structures with a SiN passivation relax slowest; the SiC passivated structures relax at an intermediate rate, while unpassivated lines relax quickly. This trend matches the electromigration resistance of the interconnect structures. Numerical simulations show that the elastic modulus of the passivation layer has a negligible effect on the stress relaxation in the lines. Consequently, we believe that the change in stress relaxation rate is caused by variations in the diffusion coefficient between copper and the two types of passivating film. Our numerical simulations suggest that stress relaxation is caused primarily by a flux of atoms from the copper/passivation interface into bamboo grain boundaries in the copper lines. There is an additional, but slower, contribution to stress relaxation arising from atoms that migrate from the copper/passivation interface into the copper/tantalum/dielectric interface on the sidewalls of the interconnect. Simulations show that the rate of stress relaxation in the patterned lines is influenced by the diffusion coefficient of the copper/passivation interface only if diffusion along this interface is significantly slower than diffusion along grain boundaries. By fitting the results of numerical simulations to experiments, we have estimated the values of diffusion coefficients for the interfaces in our specimens at 200 °C. The values are listed in Table III.

Our results suggest that stress relaxation measurements, together with numerical simulations, provide a straightforward and direct way to determine the diffusion coefficients of interfaces in representative interconnect structures. Future developments of the technique will include stress relaxation measurements over a range of temperatures, which will also allow activation energies to be determined. ACKNOWLEDGMENT

Work at Brown University and University of Texas at Austin was supported by a gift from Intel Corporation. VI. CONCLUSIONS 1

Materials that are intended to be used as interconnects or dielectrics in integrated circuits must be tested to ensure that

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