First principle investigation of hydrogen behavior in M doped Cu$_2$O (M $=$ Na, Li and Ti)

We study the hydrogen effect on the electronic, magnetic and optical properties of Cu$_2$O in presence of different dopants (Na, Li and Ti). The electronic properties calculations show that hydrogen changes the conductivity of Cu$_2$O from p to n-type. The results show that interstitial hydrogen atom prefers to locate in the tetrahedral site in Cu$_2$O system and it decreases the band gap value of the later. The Na or Li doping Cu$_2$O preserves the p-type conductivity of Cu$_2$O, while hydrogen is the source of n-type conductivity in Na or Li doped Cu$_2$O systems. Ti doping increases the band gap value of Cu$_2$O and makes it an n-type semiconductor. Hydrogen increases the optical transmittance of M doped Cu$_2$O.