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Abstract Heterogeneous integration of dissimilar crystalline materials has recently attracted considerable attention due to its potential for high-performance multifunctional electronic and photonic devices. The conventional method for fabricating heterostructures is by heteroepitaxy, in which epitaxy is performed on crystallographically different materials. However, epitaxial limitations in monolithic growth of dissimilar materials prevent implementation of high quality heterostructures, such as complex-oxides on conventional semiconductor platforms (Si, III-V and III-N). In this work, we demonstrate gallium nitride (GaN) high-electron-mobility transistors with crystalline complex-oxide material enabled by heterogeneous integration through epitaxial lift-off and direct stacking. We successfully integrate high-κ complex-oxide SrTiO3 in freestanding membrane form with GaN heterostructure via a simple transfer process as the gate oxide. The fabricated device shows steep subthreshold swing close to the Boltzmann limit, along with negligible hysteresis and low dynamic on-resistance, indicating very low defect density between the SrTiO3 gate oxide and GaN heterostructure. Our results show that heterogeneous integration through direct material stacking is a promising route towards fabricating functional heterostructures not possible by conventional epitaxy.