A layer of polymer material is applied on a peripheral region of at least one of the two substrates to be bonded prior to bonding. The bonded structure formed thereby includes a first substrate, a second substrate in direct contact with the first substrate, and a ring of the polymer material in direct contact with the first substrate at a first interface and in direct contact with the second substrate. The ring of polymer material laterally surrounds and seals the interface at which the first substrate contacts the second substrate. A ring-shaped cavity can be formed within the polymeric ring. Alternately, the first interface and the second interface can be contiguous without a ring-shaped cavity between the first and second substrates.