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].