Adam Wikström

ANALYSIS OF AVERAGE THERMAL STRESSES IN PASSIVATED METAL INTERCONNECTS

Volume-averaged thermal stresses in passivated metal interconnects on Si substrates are derived for situations where the thickness to width ratio of the interconnect lines is “small” or “large.” The analysis provides different components of volume-averaged stresses for the most general case of thermal and elastic anisotropy in the passivation layer, the interconnect line, and the substrate. It is shown that the theoretical predictions, particularly those for the hydrostatic stresses, are in agreement with detailed finite element calculations for a wide range of line and passivation geometries of practical interest. The theoretical predictions of average hydrostatic stresses are also found to be in reasonable agreement with available experimental results for thermal stresses derived from x-ray diffraction measurements on passivated Cu lines. The present theoretical results are shown to be far more accurate than prior stress analyses for periodic passivated lines based on Eshelby’s theory of inclusions.

Thermoelastic analysis of periodic thin lines deposited on a substrate

Abstract Thermoelastic stresses and curvatures arising from patterned thin lines on initially flat isotropic substrates are analyzed. A connection is made between substrates with patterned lines and laminated anisotropic composites containing transverse matrix cracks. Using this analogy along with anisotropic plate theories, approximate analytical expressions are derived for volume-averaged stresses as well as curvatures along and normal to the lines, for any thickness, width and spacing of the lines. The predictions of the analysis are shown to compare favorably with finite element simulations of stresses and curvatures for Si substrates with Al, Cu or SiO 2 lines. The predictions also match prior experimental measurements of curvatures along and normal to patterned SiO 2 lines on Si wafers, and further capture the general experimental trends reported previously for curvature evolutions in Si wafers with Al lines. The model presented here thus provides a very convenient and simple analytical tool for extracting stresses in thin lines on substrates from a knowledge of experimentally determined film stress, thereby circumventing the need for detailed computations for a wide range of unpassivated line geometries of interest in microelectronic applications.

Anisotropy and texture in thin copper films—an elasto-plastic analysis

The role of elastic anisotropy on the stress inhomogeneity and effective behavior of columnar grained textured Cu thin films have been analyzed within a continuum framework. The analysis is based on a three-dimensional model of a film/substrate system. The film exhibits a fiber texture with (111), (001) and randomly oriented grains. Mainly two load cases have been considered. Biaxial loading of a film deposited on a silicon substrate and tensile loading of a film deposited on a polyimide substrate. The stress distributions in the (111) and (001) grains were generally found to be very different when subjected to biaxial loading and quite similar when subjected to tensile loading. When plastic behavior is invoked, a structural hardening effect is observed. The plastic behavior differs significantly between biaxial and tensile cyclic loading respectively. A new orientation dependent hardening law is proposed. This hardening law causes the plastic hardening behavior to be orientation dependent and scale with elastic anisotropy. The newly proposed hardening law is demonstrated on a film with small grain aspect ratio.

Stresses in thin films and interconnect lines

The mechanical behavior of thin films and interconnect lines is investigated. Firstly, theoretical models of thermal stress evolution in thin films and passivated or unpassivated lines are considered. Secondly, the effect of texture in a copper thin film with a columnar grain structure is studied from a theoretical point of view. The film consists of three different constituents with (111), (100) and randomly oriented texture. Global properties as well as local stress distributions are considered in detail within a thermoelastic framework. The results are in qualitative agreement with available experimental results. Implications with regards to plastic behavior are briefly discussed. Finally, the potential of the curvature measurement technique for experimental stress evaluation in thin films is considered for initially flat and curved substrate/film systems.

Stresses in passivated lines from curvature measurements

Abstract An exact expression is presented which enables the determination of volume-averaged thermoelastic stresses in passivated lines based on information from curvature measurements. This method, which is valid for lines of arbitrary in-plane shape, requires knowledge of the material properties of the line and passivation. The sensitivity of the line stress estimates to uncertainties in curvature data for a typical line/passivation geometry is investigated in detail. It is concluded that the present approach is well suited for Al or Cu lines embedded in SiO 2 passivation. In the particular case of coinciding shear moduli or Poisson ratios for the line and passivation, it is shown that the method breaks down and only certain linear combinations of the line stress components may be determined.

Onset of plastic yielding in thin metal lines deposited on substrates

A thermoelastic analysis of patterned lines on substrates to determine the critical conditions for the onset of yielding is investigated, which provides a new analytical tool for determining the yield properties of lines relative to those of films of the same thickness. Verification of the theory is provided with the aid of finite element simulations, and a method for experimentally extracting the plastic properties of lines is also outlined.

Thermoelastic analysis of thin lines on substrates

General equations which are valid for arbitrary anisotropy and line/film geometry are first formulated in a systematic way. It shown that except for the thin film geometry, additional equations are required for determination of average stresses in the various constituents. The missing information is quantified in terms of the number of constituents in the line/film. A review of theoretical models which complement the general equations for unpassivated and passivated lines is provided. Some comparisons to finite element calculations are presented. Models used for interpretation of X-ray and curvature measurements in terms of average constituent stresses are also discussed. Finally some results are presented in relation to effects of an initial substrate curvature as well as consequences of plastic deformation.

Passivated Interconnect Lines: Thermomechanical Analysis and Curvature Measurements

It is well known that curvature measurements may be used to obtain volume averaged stresses in thin continuous films and unpassivated lines without knowledge of the material properties of the film or lines. However, recently a method was presented which makes it possible to use curvature measurements also for the determination of volume averaged stresses in passivated lines. Since the problem is statically indeterminate the method requires knowledge of the material properties of the lines and passivation. The sensitivity of the method to uncertainties in material properties and curvature data is here investigated by utilizing the finite element method for anisotropic Cu or Al lines embedded in SiO 2 passivation. Furthermore, the method is extended to cover the case of different stress-free temperatures for the lines and passivation respectively.

Stresses in Thin Coatings from Curvature Measurements on Non-Planar Substrates

Mechanical elastic and inelastic properties of thin coatings are often studied by means of the curvature measurement technique in combination with the Stoney formula. It is then implicitly assumed ...