Here, we explore, on grounds of density-functional theory (DFT -for details see Methods section and Supporting Information (SI)), the chemical interaction and mobility of both H and H + species in the layered materials of h-BN, MoS 2 , and graphite. We show that both H + and H can be accommodated in h-BN and MoS 2 , and that diffusion is possible via a low-energy transport mechanism that is assisted by the rigid-layer shear modes, which have low frequency due to weak interlayer-interactions 10 (these modes have been observed in the Raman spectra at 52.5 cm - 1 and 33.7 cm -1 for h-BN and MoS 2 , 11 respectively). For both H and H + transport follows a zigzag hopping motion between adjacent layers. Diffusion in MoS 2 has a lower barrier than in h-BN, however, sulfur vacancies -a typical defect in MoS 2 -act as hydrogen traps and suppress the transport. In graphite, protons are immediately neutralized to H atoms, which themselves are almost immobile once bound to the carbon atoms. Results and Discussions