An integrated circuit containing an n-channel finFET and a p-channel finFET has a dielectric layer over a silicon substrate. The fins of the finFETs have semiconductor materials with higher mobilities than silicon. A fin of the n-channel finFET is on a first silicon-germanium buffer in a first trench through the dielectric layer on the substrate. A fin of the p-channel finFET is on a second silicon-germanium buffer in a second trench through the dielectric layer on the substrate. The fins extend at least 10 nanometers above the dielectric layer. The fins are formed by epitaxial growth on the silicon-germanium buffers in the trenches in the dielectric layer, followed by CMP planarization down to the dielectric layer. The dielectric layer is recessed to expose the fins. The fins may be formed concurrently or separately.