The crystalline structures and Brillouin zones of rutile and anatase . The crystal structures are optimized via the HSE approach with: a = 4.737 Å; c = 3.186 Å for rutile SnO2 (r-SnO2), a = 4.592 Å; c = 2.948 Å for rutile TiO2 (r-TiO2), a = 3.975 Å; c = 10.203 Å for anatase SnO2 (a-SnO2), and a = 3.766 Å; c = 9.636 Å for anatase TiO2 (a-TiO2). The optimization reveals that and internal lattice parameter u is fairly similar for SnO2 and TiO2: u = 0.306 for r-SnO2 and 0.305 for r-TiO2, whereas u = 0.800 for a-SnO2 and 0.794 for a-TiO2.
Electronic structures of r-SnO2, r-TiO2, a-SnO2, and a-TiO2 along two main symmetry directions (100) = ⊥ and (001) = ||. Solid lines and circles represent the HSE and GW0 energies, respectively. Energies refer to the VBM. Spin-orbit interaction is included.
Close-up of the topmost VBs at the VBM from Fig. 2 .
Atom and angular resolved DOS near the band-edges of r-SnO2, r-TiO2, a-SnO2, and a-TiO2, using the HSE functional and including a 0.03 eV Lorentzian broadening. The DOS has been scaled by (2l + 1)−1 for better visibility. The main difference between SnO2 and TiO2 is the strong cation 3d-character of the lowest CBs of TiO2.
The imaginary part ε 2(ω) of the dielectric function from the HSE calculations of r-SnO2, r-TiO2, a-SnO2, and a-TiO2. The spectra are divided into transverse (⊥) and longitudinal (||) components, including a 0.03 eV Lorentzian broadening.
The real part ε 1(ω) of the dielectric function obtained from the Kramers-Kronig transformation.
The phonon parts of real dielectric functions simulated from the Lorentz model and Kramers-Heisenberg formula with multiphonon contributions in the limit of zero damping parameter ( ). The spectra are divided into transverse and longitudinal components, with the corresponding LO-TO splitting calculated using non-analytical correction.
Calculated Γ-point band energies En (Γ) of rutile SnO2(r-SnO2), rutile TiO2(r-TiO2), anatase SnO2(a-SnO2), and anatase TiO2(a-TiO2) in units of eV and referred to the VBM; cf. Figs. 2 and 3 . Spin-orbit coupling is included. The c 1 refers to the bottommost of CB, v 1, v 2, and v 3 refer to the uppermost, second, and third VB, respectively; see Fig. 3 .
The calculated effective electron and hole masses and at CBM and VBM together with the available experimental values. The corresponding polaron masses mp are also listed. The spin-orbit coupling is included.
Calculated high frequency and static dielectric constants ε∞ and ε0 together with the experimental values. The || and represent the x-y plane and z-direction, respectively.
The calculated infrared-active phonon frequencies of SnO2 and TiO2 at Г point using finite displacement method with non-analytical correction together with the experimental values. The frequencies are in units of meV. , where ω LO, j and ω TO, j are the LO and TO frequencies of the jth phonon mode (… represent there is no corresponding vibration modes).
The Born effective charge of SnO2 and TiO2 calculated from the derivative of the force on nuclei by a homogeneous effective electric field at the zero atomic displacement. The x, y, z directions along the a, b, and c crystal axis in the conventional cell of rutile and anatase. For rutile structure, only is presented ( ).
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