4.2 Stress and its origins in heterostructure
4.2.1 Intrinsic stress and its origins
determined by the direction of it, if it suffers tensile stress, which the direction is the same with that of the direction of the coordinate, then the sign is positive and the wafer bending is concave with positive curvature value. Conversely, when the film suffers compressive stress, the wafer bowing is convex with minus curvature value, as represented in Fig. 4.2.
of the underlying layer (𝑎𝑎𝑠𝑠) and is forced to be the same as𝑎𝑎𝑠𝑠. Then the first layers grow pseudomorphically. The misfit strain 𝜖𝜖𝑚𝑚 is defined as Eq. (1-1) and is repeated here
𝜖𝜖𝑚𝑚= 𝑎𝑎𝑠𝑠0𝑎𝑎−𝑎𝑎𝑓𝑓0
𝑓𝑓0 (4-10) with 𝑎𝑎𝑠𝑠0and 𝑎𝑎𝑓𝑓0 being the neutral value of the lattice of substrate and film which are stress-free. In most cases, substrate is assumed to be stiff and 𝑎𝑎𝑠𝑠 doesn’t change. The misfit stress 𝜎𝜎𝑚𝑚 caused by misfit strain can be calculated as
𝜎𝜎𝑚𝑚 = 𝑀𝑀𝑓𝑓∙ 𝜖𝜖𝑚𝑚 = 𝑀𝑀𝑓𝑓∙𝑎𝑎𝑠𝑠0𝑎𝑎−𝑎𝑎𝑓𝑓0
𝑓𝑓0 (4-11) with 𝑀𝑀 being the biaxial elastic modulus of the film. The magnitude of 𝜎𝜎𝑚𝑚 can be 1 ~ 10 GPa, which is very huge compared with the contributions of other mechanisms and is the most dominant and influential one [8, 9]. However, the misfit stress calculated by Eq. (4-10) is only the ideal case that the film is completely strained, which is almost impossible. In fact, as introduced in chapter 3, there are many strain relaxation mechanisms to relief the misfit strain. The true strain after relaxation is given by
𝜖𝜖 = 𝑎𝑎𝑓𝑓𝑎𝑎−𝑎𝑎𝑓𝑓0
𝑓𝑓0 (4-12) with 𝑎𝑎𝑓𝑓 being the relaxed lattice constant. Even for the perfect pseudomorphical growth, it would relax by forming misfit dislocations if the thickness is beyond a critical value which is usually several or tens of nanometers [10, 11]. The relaxation amount 𝜍𝜍 is defined as
𝜍𝜍= 𝑎𝑎𝑎𝑎𝑓𝑓−𝑎𝑎𝑠𝑠
𝑓𝑓0−𝑎𝑎𝑠𝑠0× 100% (4-13) with 𝑎𝑎𝑓𝑓0 and 𝑎𝑎𝑠𝑠0 being the neutral lattice constant of the film and substrate respectively, and 𝑎𝑎𝑓𝑓 and 𝑎𝑎𝑠𝑠 being the strained lattice constant of the film and substrate under the growth conditions respectively. As it is assumed that the substrate is stiff and 𝑎𝑎𝑠𝑠0 doesn’t change, so 𝑎𝑎𝑠𝑠 =𝑎𝑎𝑠𝑠0. Misfit stress is the most important intrinsic stress that to be considered in this study. The sign of stress follows the conventional regulation that if the film suffers tensile stress (𝑎𝑎𝑠𝑠0− 𝑎𝑎𝑓𝑓0> 0) it is positive and if the stress is compressive (𝑎𝑎𝑠𝑠0− 𝑎𝑎𝑓𝑓0 < 0) it is negative. So does the curvature.
• Solid state reactions and/or diffusion
This mechanism refers to the cases that there is solid phase reaction or specie diffusion at the interface between a substrate and the film on it [8]. When there is solid phase reaction, new compound is formed at the interface. After the formation of new compound, the diffusion of reactant to the other side continues. Both processes can be accompanied by the raise of stress. Stress can also be induced if there is only diffusion from one side to the other.
For example, the oxygen incorporation in Al substrate diffuses to the clean Al film on it. The diffused atom species can incorporate into the crystal lattice of the film and leads to lattice constant variation and then induce stress at the interface. For the particular case here, the react between gallium and silicon also leads to the stress relief during the growth by forming voids in silicon substrate and Ga-Si alloy spots in the film and destroying the epitaxial
87
structure of nitrides on silicon. These large size (1 ~ 100 um) structural defects keep the growth from forming an entire film.
4.2.1.2 Intrinsic stress within the films
This is the most complicated part of the origins of stress since it relates to the microstructure of the film during the processes of growth.
• Small-angle grain boundaries
This mechanism exists in polycrystalline and amorphous films which consist of numerous randomly oriented grains with small angles and size from several nanometers to micrometers and contain a great concentration of grain boundaries. There are gaps between these grains and the interatomic forces tend to close them and then result in tensile stress in neighboring crystallites [8].
• Domain walls or coalescence of grain boundaries
This only emerges in epitaxial films grown in Volmer-Weber mode. The as contacted islands have larger surface energy 𝛾𝛾𝑠𝑠 (Fig. 4.3a) than the interface energy 𝛾𝛾𝑆𝑆 (Fig. 4.3b). The system total free energy can decrease by forming interface and annihilating free surface. At the beginning of coalescence, similar to the mechanism above of small-angle grain boundaries, the interatomic forces between neighboring islands tends to close their gap and this leads to a short period of tensile stress. Next, during coalescence and increase of thickness, atoms arrive at and fill the grain boundaries or domain walls, resulting in compressive stress. The later stage is similar to the process of insertion of excess atoms in grain boundaries.
• Recrystallization processes
Stress may emerges during the processes of recrystallization, during growth, annealing and after growth, especially in the films with inferior quality. Recrystallization brings better crystal quality. In some cases, recrystallization may increase the density of the film and cause shrinkage of it. Since the film is attached to the substrate tightly and the substrate tends to keep the film from shrinkage and then tensile stress in the film is generated [8, 12].
• Annihilation of excess vacancies
In most cases, there is a great deal of vacancies in crystals. They can annihilate in the grains, at the boundaries, at the free surfaces and the surfaces of internal cavities. No stress is caused if they are annihilated at the surfaces while a tensile stress is built up if they are
Fig. 4.3 Compressive stress generation during coalescence of crystal islands [12].
(a) (b)
88
annihilated in the grains or at the boundaries for that their annihilation generates gaps and the grains tends to move to close the gap [12].
• Impurities
Impurities can cause stress even in homogenous epitaxial growth by doping the film. This mechanism has been mentioned in the introduction of the mechanism of solid state diffusion at the interface in the last subsection. Impurities or dopants can replace the lattice site of some original atom species. Due to the difference of the radius between the impurity atoms and the atoms of the epitaxial film, true lattice constant of the film can be changed [8].
There are some other mechanisms that can induce intrinsic stress within the film, such as the lattice expansion in capillarity effect, the capillarity stress and so on [8].