2. Computational methods
The geometric structure, the electronic structure, and the optical properties of the TiOS doped with ten 4d TM atoms Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd are calculated using the plane wave method based on the density functional theory (DFT). First, all the TiOS models doped in different doping manners are optimized, and the optimal doping manners of all TM atoms can be determined according to the impurity formation energy. Second, the band structures, the density of states, and the optical absorption coefficients of all the best doping models are calculated. All the calculations are performed with the Cambridge Serial Total Energy Package (CASTEP) [29]. The exchange-correlation potential is described by the Perdew-Burke-Ernzerhof (PBE) functional of the generalized gradient approximation (GGA) [30]. The cutoff energy for the plane wave basis set is set to be 340 eV. The Monkhorst-Pack scheme k-point grid sampling for the reduced Brillouin zone is set as 2 × 4 × 1, and the convergence criteria for the self-consistent field (SCF) is set to 1.0 × 10−6 eV/atom. All the atom coordinates are fully optimized until the forces on every atom are smaller than 0.03 eV/Å. The interaction between the valence electrons and the ionic core is described by the ultrasoft pseudopotential [31].
The formation energy is defined as
, (1)
where E (M ) is the energy of an isolated TM atom, andE (TiO2 ) [E (M-TiO2 )] is the total energy of the pure [TM-atom-doped] TiOS. For convenience, all of the optimal doping models can be denoted as Ti24MO48with M representing the TM atoms Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd.
As widely recognized, the transfer rate of charge carriers can play a vital role in determining the photoactivity of a semiconductor material [32,33], and can be defined as [32,34]
\(v=\hslash k/m^{*}\), (2)
where ħ is the reduced Planck constant, k is the wave vector, and m* is the effective mass of charge carrier. The effective masses of electrons (me* ) and holes (mh* ) can be obtained by fitting the energy band around the valence band maximum (VBM) and the conduction band minimum (CBM) along a specific direction in the reciprocal space, respectively. It should be noticed that the region used for parabolic fitting is kept in a very small interval to ensure the validity of the parabolic approximation. In addition, to clarify the difference between the effective masses of electrons and holes, an effective mass ratiorhe is introduced as
\(r_{\text{he}}=\frac{{m_{h}}^{*}}{{m_{e}}^{*}}\). (3)
In general, a larger rhe value indicatesmh* >me* , resulting in a larger difference of the carrier velocites and a slower recombination rate of the photogenerated electron-hole pairs [32].