The invention provides methods for direct growth of low noise, atomically thin freestanding membranes of two-dimensional monocrystalline or polycrystalline materials, such as transition metal chalcogenides including molybdenum disulfide. The freestanding membranes are directly grown over an aperture by reacting two precursors in a chemical vapor deposition process carried out at atmospheric pressure. Membrane growth is preferentially over apertures in a thin sheet of solid state material. The resulting membranes are one or a few atomic layers thick and essentially free of defects. The membranes are useful for sequencing of biopolymers through nanopores.