Structures for an avalanche photodetector and methods of forming a structure for an avalanche photodetector. The structure includes a first semiconductor layer having a first portion and a second portion, and a second semiconductor layer stacked in a vertical direction with the first semiconductor layer. The first portion of the first semiconductor layer defines a multiplication region of the avalanche photodetector, and the second semiconductor layer defines an absorption region of the avalanche photodetector. The structure further includes a charge sheet in the second portion of the first semiconductor layer. The charge sheet has a thickness that varies with position in a horizontal plane, and the charge sheet is positioned in the vertical direction between the second semiconductor layer and the first portion of the first semiconductor layer.