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= Residual stress in thin films =

Large residual stress (stress persist even after the external force is removed) can develop during the growth of thin films. Residual stresses of some hundred MPa or even few GPa can be found in thin films, for instance in sputtered TiN films [1]. Such strong pressures are higher than the ones at the bottom of the Mariana Trench, the deepest natural trench in the world (~110 MPa, 1,000 atm). However, the hydrostatic pressure is isotropic while the mechanical stresses in thin films are strongly anisotropic [2]. A large amount of stored stress in a film/substrate system, can lead to crack generation and propagation, and a subsequent delamination of the film. An interesting example is the formation of telephone cord blisters, observed in many compressed film-substrate systems and caused by biaxial compressive residual stresses [3]. Even though residual stresses can bring deleterious effects, they can also be used to improve the mechanical properties of materials. For example, shot peening method causes compressive residual stress on superficial layers which is used to increase both work-hardening (higher yield strength but lower ductility) and resistance to fatigue. These characteristics are used in toughened glass and turbine fans, improving their resistance to corrosion and cracking.

Origin of residual stresses
In general, the deposition of a film is a non-equilibrium process in which the species do not have enough time to find the most stable state (minimize free energy). Since the atoms are not positioned in their fully relaxed position, stresses are created during the growth of a film. The difficulty on understanding the stress evolution relies on the fact that this process depends on many parameters such as growth rate, material type, grain size, morphology, temperature, substrate, etc. [4] Figure 1 shows the three main sources of residual stresses in films.

Lattice mismatch stress
Also known as epitaxial stress, it arises when the crystal lattices of the film and the substrate line up perfectly (if luckily enough). If the crystal structure is maintained, the misfit strain (εmf)

$$

\epsilon_{xx}=\epsilon_{yy} = {{a_{s}-a_{f}} \over a_{f}}=\epsilon_{mf}$$

As observed in Fig. 1, there is an in-plane lattice expansion and an out-of plane lattice contraction in the film. To calculate the misfit stress, the generalized form of Hooke’s law needs to be employed. This equation links the two 2nd rank tensors of interest (σij-stress and εij-strain) by a 4th rank tensor cijkl-elastic stiffness (81 components) which due to symmetry can be reduced to 36 components. The number of independent components of the stiffness matrix will depend on the crystal structure of interest. For instance, cubic has three independent components (highest symmetry), hexagonal five, triclinic 21 (lowest symmetry). [5] It is important to note that the misfit strain holds exclusively when the film is sufficiently thin. When the film thickness increases, misfit dislocation at the interface reduce the stress (relaxation) in the film. Matthews equation introduces the idea of critical thickness in which it becomes favorable to form misfit dislocations. This is an important parameter in device fabrication. For instance, due to lattice mismatch between GaN and GaInN, a partial/full relaxation is observed for thicknesses higher than the critical thickness. This creates a misfit dislocations (defects) that induce nonradiative recombination centers [6], not desirable for LED technology.

Thermal residual stress
Thermal residual stress is due to the difference in the thermal expansion coefficient Δα of the film and the substrate. For a deposition temperature Tdep, the thermal stress is proportional to:

$$

\sigma_{T}\sim(T_{dep}-T)\Delta\alpha-\epsilon_{pl}$$

The plastic strain εpl arises for large enough Δα and ΔT if the film stress exceeds the yield strength; in this case the elastic strain will vanish and only εpl remains. This stress may cause problems for instance in SiN films which are commonly used as passivation layers. Cracking may occur in SiN film over aluminum pads. [7]

Intrinsic or growth stress
During deposition of the film, redistribution of matter will result in intrinsic stresses. Different mechanisms can cause intrinsic stresses such as evolution of grain size, surface energy (important for very thin films), grain growth rate, recrystallization, grain boundary relaxation, crystallite coalescence, formation of non-equilibrium phases (phase transition), defect incorporation into the film, etc. [2, 4, 8] Due to the different contributions of the intrinsic stresses, both the magnitude and sign of the stresses can be modified during growth. Because of its high mechanical strength, SiC films can be used in Micro-Electro-Mechanical Systems (MEMS) structure. However, an unwanted deformation in the film would alters the resonant frequency, then it is necessary to control the intrinsic residual stresses during growth. The intrinsic stress of amorphous SiC:H films deposited by PECVD has been extensively studied. It was observed that the as-deposited films show an increase in compressive residual stress (negative sign) with the increase in deposition temperature. Post-annealing process reduces the residual compressive stress with increasing temperature and further causes an overall tensile stress (positive sign). This last result is correlated with a loss of hydrogen in the film (27 a.t. % at 250 °C to 12 a.t. % at 650 °C). Also, it was observed that an increase in the annealing time causes the film to relax up to a compressive stress of –50 MPa [9]. In general, many trends have been identified, but also many exceptions [4].

Measuring residual stress in thin films
A wide range of characterization techniques, destructive or non-destructive, may be used to quantify the residual stress in thin films. Since stress (and residual stress) is an intensive property, characterization methods focus on a proper measurement of strain (extensive). For thin films different characterization methods have been used. One frequent way to determine stresses is by measuring the curvature generated by a coating or a film on the substrate either by contact methods (profilometry) or without direct contact (video, laser scanning). Other methods include X-ray diffraction (XRD), neutron scattering, electron diffraction, photoacoustic emission, and magnetoacoustic emission. [2, 10, 11]