Strain energy due to splitting forces as a basis for bonding criterion in wafer bonding

• The total strain energy of a bilayer structure and the SSE always consist of two integral parts: the strain energy due to bending moment and due to axial force in the ratio 3:1.
• The strain energy due to splitting forces has been isolated from the total strain energy of bilayer.
• The strain energy due to splitting forces is proposed as the specific strain energy (SSE) for the bonding criterion.
• The new SSE strongly depends on the thickness ratio of the layers and will even equal zero at a thickness ratio known as the point of equivalence. At the point of equivalence, even strong internal stresses do not affect the bonding strength of bilayer.
• The point of equivalence has been calculated for five structures: Si/Al2O3, Si/GaAs, Si/LiNbO3, Si/SiC, and Si/InP.

Direct wafer bonding is an important technology in sensor manufacturing, but bonding strength between different materials is still in need of improvement due to thermally induced internal stresses. To estimate the negative influence of the internal stresses, a bonding criterion has been established that compares the surface energy, which dissipates as the bond is formed, to the specific strain energy, which is stored in bilayers structure due to internal stresses. In the literature, general practice assumes that the specific strain energy is the strain energy of pure bending. However, from the physical standpoint, the strain energy due to pure bending represents neither the total strain energy nor the strain energy due to the splitting forces.We have calculated the exact total strain energy of a curved bilayer and its two components: the strain energy due to the forces parallel and perpendicular to the interface. The strain energy of the forces perpendicular to the interface (splitting forces) has been proposed as the specific strain energy for the bonding criterion. The new specific strain energy strongly depends on the thickness ratio of the layers and can even equal zero at a certain thickness ratio called the point of equivalence. In turn, the point of equivalence does not depend on the total thickness of the bilayer structure. The point of equivalence, where internals stresses do not affect the bonding strength, was calculated for five structures: Si/Al2O3, Si/GaAs, Si/LiNbO3, Si/SiC, and Si/InP.The derived result has considerable importance for applications because it demonstrates how to avoid completely the negative influence of even strong internal stresses on the bonding strength.