A Sparse Merkle tree is a type of Merkle tree, but it is much easier to prove non-membership in a sparse Merkle tree than in an arbitrary Merkle tree. For an explanation of sparse Merkle trees, see here.
This repo provides a circuit implementation of the Sparse Merkle Tree with Halo2's PLONKish arithmetization and Poseidon hash primitives.
The implementation of the native smt data structure and the Path chip is adapted from Webb and Arkworks: https://github.com/webb-tools/arkworks-gadgets