{\it Ab initio} molecular dynamics of a proton in amorphous SiO$_2$ illustrating the hopping mechanism

The scaling of metal-oxide-semiconductor devices to smaller dimensions is a major issue in current silicon technology. In order to understand the role of hydrogen at silicon-oxide interfaces, we here investigate charged states of hydrogen in amorphous SiO$_2$ ($a$-SiO$_2$) using first-principles calculations (DFT-GGA). We first show that the formation energies of H$^0$, H$^+$ and H$^-$ in $a$-SiO$_2$ are essentially equivalent to those in $\alpha$-quartz. In particular, the H$^+$ and H$^-$ species are always more stable than their neutral counterpart. Then, we focus on the basic diffusion mechanism of the proton in $a$-SiO$_2$. Our molecular dynamics simulations show that the proton hops between O atoms. The hopping does not occur between first O neighbors connected through the network, but takes place across rings when the O--O distance is about 2.3 \AA. The hopping process is favored by the thermal vibrations of the O atoms.