We first concentrate on h-BN: the low-energy path for H and H + diffusing through h-BN are shown in Figure -b and Figure , respectively. The estimated free energy is 0.46 eV for H + and 0.08 eV for H, which is perfectly consistent with the experimental data. In our simulations, H + (H atom) is initially bound to the nitrogen atom N1 (boron atom B1) in one layer (black; Figure and), which are the preferential binding sites. Layered materials, such as h-BN, exhibit low-energy shearing modes. Once the shearing brings a N atom of the adjacent layer to the vicinity of the proton, there is a high probability that proton jumps to the closest N atom (N2, red) in the adjacent layer. The next jump (N2 to N3, green) then goes back to the first layer, and so forth, resulting in a zigzag transport path between layers. Thus, after each two successive jumps, the proton can be displaced by 2.5 Å, or the shortest intralayer N-N distance.