Local electronic structure of dilute hydrogen in $\beta$-MnO$_{2}$

The electronic and magnetic states of $\beta$-MnO$_{2}$ in terms of hydrogen impurities have been investigated by muon spin rotation ($\mu$SR) technique combined with density-functional theory (DFT) calculations for muon as pseudo-hydrogen. We found that 85% of implanted muons are localized in the oxygen channels of the rutile structure and behave as interstitial protons (Mu$^{+})$ except those (7.6%) forming a charge-neutral state (Mu$^{0})$ at 2.3 K, which indicates that interstitial hydrogen acts as a shallow donor within less than 0.1 meV of ionization energy. The residual 15% of muons are attributed to those related to lattice imperfection as Mn vacancies. Detailed analyses combined with DFT approach suggested that the muon is localized at the center of the oxygen channel due to its large zero-point vibration energy.