In order to compensate the insufficient conductance of heterojunction thin films, transparent conductive oxides (TCO) have been used for decades in both-sides contacted crystalline silicon heterojunction (SHJ) solar cells to provide lateral conduction for efficient carrier collection. In this work, we substitute the TCO layers by utilizing the lateral conduction of c-Si absorber, thereby enabling a TCO-free design. A series resistance of 0.32 Ωcm2 and a fill factor of 80.7% were measured for a TCO-free back-junction SHJ solar cell with a conventional finger pitch of 1.8 mm, thereby proving that relying on lateral conduction in the c-Si bulk is compatible with low series resistances. Achieving high efficiencies in SHJ solar cells with TCO-free front contacts requires suppressing deterioration of the passivation quality induced by direct metal-a-Si:H contacts and in-diffusion of metal into the a-Si:H layer. We show that an ozone treatment at the a-Si:H/metal interface suppresses the metal diffusion and improves the passivation without increasing the contact resistivity. SHJ solar cells with TCO-free front contacts and ozone treatment achieve efficiencies of > 22%.