Effects of Nb concentration on Nb-doped anatase TiO₂: DFT + U calculations

The optimized lattice parameters, average bond lengths, and differences in volume at various Nb concentrations are listed in table 1. The lattice parameters of pure anatase TiO₂ are calculated as a = b = 3.799 Å, c = 9.708 Å, close to the earlier theoretical works and experimental values [28–31], which indicates that the applied calculation methods are rational and the obtained results are reliable. In the crystal structure shown in figure 1(a), each Ti atom bonds to the nearest four oxygen horizontally and to the nearest two vertically. Average bond lengths in horizontal and vertical direction are represented as Ti–O^1st and Ti–O^2nd, respectively. With the increase of Nb-doping concentration, the volume increases gradually. This may be attributed to the difference in ion radii between Nb⁵⁺ (0.62 Å) and Ti⁴⁺ (0.56 Å) [32]. The bond length in the crystal varies accordingly after Nb atom replace Ti atom in an alternative way. In the crystal structure with the same Nb-doping concentration, the lengths of Nb–O^1st and Nb–O^2nd are longer than those of Ti–O^1st and Ti–O^2nd, respectively. As the number of substituted Ti atoms increases, the lengths of Ti–O^1st, Ti–O^2nd, and Nb–O^1st become longer, eventually causing the overall volume to expand.

In order to further observe the changes of internal atomic position due to Nb-doping, the atomic positions and the bonds on the (010) plane are presented in figure 2. The corresponding bond lengths of pure TiO₂ in figure 2(a) are set as a contrast. After Nb atom replace Ti atom, the Nb–O bond lengths along the [100] direction and most of the Nb–O bond lengths along the [001] direction are longer than those of pure TiO₂. When the Nb-doping concentration is 14.58%, the bond lengths in these two directions appear the maximum value (2.039 Å and 2.253 Å), and the local lattice deteriorates the most seriously, which is consistent with the volume difference in table 1.

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Defect formation energy can be applied to represent the stability of the doping system. The smaller the formation energy is, the more stable the doping system is. The defect formation energy is calculated according to the following formula:

$$\begin{eqnarray}&&{E}_{form}={E}_{total}\left(\mathrm{Nb} \mbox{-} \mathrm{doped}\right)-{E}_{total}\left({\rm{pure}}\right)-n{\mu }_{{\rm{Nb}}}+n{\mu }_{{\rm{Ti}}}\end{eqnarray} \tag{ 1 }$$

Where ${E}_{total}\left(\mathrm{Nb} \mbox{-} \mathrm{doped}\right)$ and ${E}_{total}\left({\rm{pure}}\right)$ represent the total energy of the Nb-doped TiO₂ supercell and pure TiO₂ supercell, respectively; n is the number of substitutional Nb atoms; ${\mu }_{{\rm{Nb}}}$ and ${\mu }_{{\rm{Ti}}}$ represent the chemical potentials of the Nb and Ti atom, respectively. ${\mu }_{{\rm{O}}}$ is used to represent the chemical potentials of the O atom. ${\mu }_{{\rm{Ti}}}$ and ${\mu }_{{\rm{O}}}$ are influenced by the growing environment, and they both satisfy the thermodynamic stability conditions of ${\mu }_{{{\rm{TiO}}}_{2}}={\mu }_{{\rm{Ti}}}+2{\mu }_{{\rm{O}}}.$ Under the O-rich growth condition, ${\mu }_{{\rm{O}}}={\mu }_{{{\rm{O}}}_{2}}/2$ (${\mu }_{{{\rm{O}}}_{2}}$ is the ground state energy of an O₂ molecule) and ${\mu }_{{\rm{Ti}}}={\mu }_{{{\rm{TiO}}}_{2}}-2{\mu }_{{\rm{O}}}.$ Under Ti-rich condition, ${\mu }_{{\rm{Ti}}}$ is the energy of one Ti atom in bulk Ti, and ${\mu }_{{\rm{O}}}=({\mu }_{{{\rm{TiO}}}_{2}}-{\mu }_{{\rm{Ti}}})/2.$ In addition, ${\mu }_{{\rm{Nb}}}$ is the energy of one Nb atom in bulk Nb.

According to Formula (1), the formation energies of Nb-doped TiO₂ under O-rich and Ti-rich conditions are calculated individually, as shown in table 2. The formation energy in O-rich environment is lower than that in Ti-rich environment, gradually decreases with the increase of Nb concentration, indicating that it is easier for Nb atoms to enter the position of TiO₂ lattice to replace Ti atoms, and higher concentrations of Nb facilitate the synthesis of Nb-doped anatase TiO₂ systems. By contrast, in the Ti-rich environment, the formation energies of Nb-doped anatase TiO₂ are positive, which shows that the synthesis of the Nb-doped anatase TiO₂ systems is relatively difficult.

Ab initio molecular dynamics (AIMD) is currently a popular and expanding computational tool employed to study physical, chemical, and biological phenomena [33]. We investigated the thermal stability of the Nb-doped anatase TiO₂ using AIMD simulations, and a high temperature of 900 K was adopted to accelerate the phase-change process. The time evolutions of the average energy per atom fluctuation for Nb-doped anatase TiO₂ from 1 to 10000 fs at the temperature of 300 K and 900 K are presented in figure 3. At room temperature, the energy fluctuation of TiO₂ with Nb-doping concentration from 2.08 at.% to 14.58 at.% is very small. When the temperature is increased to 900 K, the energy fluctuation becomes larger, but the fluctuation range remains within 0.1 eV/atom [34]. The results indicate that the structures of TiO₂ with Nb-doping concentration from 2.08 at.% to 14.58 at.% are stable with respect to the thermal shock and there is no phase transitions observed to these structures.

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The calculated band structure of pure TiO₂ is plotted in figure 4(a). It is found that the pure anatase TiO₂ is a direct band gap semiconductor with a band gap of 3.197 eV at the G point, which is in good consistency with the experimental value of 3.2 eV. After Nb doping into the TiO₂ lattice, no impurity level appears in the band gap of TiO₂, but Fermi level enters into the conduction band, which shows a typical n-type metallic characteristic in electronic structure. With the increase of Nb concentration, Fermi level enters deeper into the conduction band, and the band gap value of TiO₂ decreases gradually, but the optical band gap value increases from 3.197 eV to 3.771 eV of 14.58 at.% concentration, as shown in figures 4(b)–(f) and 6. The widening of the optical band gap can be explained by Burstein-Moss effect [35], which results from the Pauli exclusion principle. After the Fermi level enters the conduction band, all states below the Fermi level are occupied states. The electrons at the top of the valence band cannot be excited to enter these occupied states due to the restriction of Pauli exclusion principle, so they can only be excited to enter the conduction band above the Fermi level, and the absorption edge transfers to a higher level of energy, as a result the optical band gap widens.

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Figure 5 shows the total density of states (TDOS) and partial density of states (PDOS) of Nb-doped TiO₂. In figure 5(a), the valence band of TiO₂ mainly consists of O 2p and Ti 3d states, and the conduction band is mainly formed from Ti 3d, O 2p states. After the replacement of Ti atom by Nb atom, the occupied states at the bottom of the conduction band facilitate the Fermi level to enter the conduction band, and hybridization of Ti 3d, Nb 4d and O 2p states in the conduction band makes the bottom of the conduction band move towards the lower energy level, and the band gap value of Nb-doped TiO₂ decreases from 3.197 eV to 1.91 eV, as shown in figures 5(b)–(f) and 6. By integrating the occupied states near Fermi level, the number of electrons entering the conduction band increases with the increase of Nb concentration, which means that the conductivity of Nb-doped TiO₂ is enhanced. In the valence band, the overlap of Ti 3d, O 2p and Nb 4d states widens the valence band, and the number of electrons in the valence band also increases. The electrons can be excited to the conduction band through absorbing sufficiently high photon energy, which is another reason for the enhancement of conductivity. However, the conductivity of Nb-doped TiO₂ does not always increase with the increase of Nb concentration. When Nb concentration is excessively high, the electronic states in the crystal will be modified by carrier-carrier and carrier-defect interactions, resulting in the scattering phenomenon [36, 37], which will reduce the hall mobility and deteriorate the conductivity. Therefore, Nb doping concentration needs to be controlled within a reasonable range considering the analysis of electrical and optical properties, which will be discussed in the later sections.

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Through the Mulliken population analysis, charge distribution and charge transfer of the material can be understood. In the pure TiO₂ supercell, one Ti atom loss 1.33 electrons, one O atom nearby Ti atom accepts 0.67 electrons, as shown in table 2. With the increase of Nb concentration, the Mulliken populations of Ti and Nb decrease gradually, while the Mulliken population of O remains basically unchanged, indicating that the incorporation of Nb reduces the number of electrons transferred from Ti and Nb atoms to O atoms. The Mulliken populations of Nb are larger than that of Ti in the same Nb concentration, which means that Nb dopant contributes more electrons to the Nb-doped TiO₂ supercell than Ti. The extra electrons are delocalized at the bottom of the conduction band and will lead to the transformation of Ti⁴⁺ to Ti³⁺ [14, 38]. In other words, the conductivity of the Nb-doped TiO₂ is attributed to the presence of Ti³⁺ ions [39].

Hall mobility, carrier concentration and conductivity have been three important parameters to characterize the electrical properties of TCO materials. Nb-doped anatase TiO₂ exhibits the n-type conducting characteristics, which mainly depends on the free electrons for conduction. The electron mobility can be expressed by the following equation [40]:

$$\begin{eqnarray}&&\mu =\displaystyle \frac{q{\tau }_{i}}{{m}_{n}^{* }}\end{eqnarray} \tag{ 2 }$$

Where q is the electronic charge, ${\tau }_{i}$ is the relaxation time, ${m}_{n}^{* }$ is electron effective mass. The relaxation time ${\tau }_{i}$ is a function of ionized impurity concentration ${N}_{i}$ and thermodynamic temperature T: ${\tau }_{i}={N}_{i}^{-1}{T}^{3/2}.$ The electron effective mass ${m}_{n}^{* }$ can be obtained by calculating the average of the lateral and longitudinal effective masses in the Nb-doped TiO₂ crystal structure, which is expressed by equation (3) [41].

$$\begin{eqnarray}&&\displaystyle \frac{3}{{m}_{n}^{* }}=\displaystyle \frac{2}{{m}_{a}^{* }}+\displaystyle \frac{1}{{m}_{c}^{* }}\end{eqnarray} \tag{ 3 }$$

Where ${m}_{a}^{* }$ is the electron effective mass in the [100] direction, ${m}_{c}^{* }$ is the electron effective mass in the [001] direction. According to the band structure, ${m}_{a}^{* }$ and ${m}_{c}^{* }$ at the bottom of the conduction band can be individually calculated by equation (4)) [42].

$$\begin{eqnarray}&&\displaystyle \frac{1}{{m}^{* }}=\displaystyle \frac{1}{{ \hbar }^{2}}\displaystyle \frac{{d}^{2}E(k)}{d{k}^{2}}\end{eqnarray} \tag{ 4 }$$

Where E(k) is the energy, k is the wave vector, ħ is the reduced Planck constant. According to equations (3) and (4), the effective mass of TiO₂ doped with various Nb concentrations is showed in table 3. The electron effective masses along the [100] and [001] direction of TiO₂ with Nb-doping concentration of 2.08 at.% are consistent with the experimental results of Hirose's research group [43]. They identified the electron effective mass along the a-axis ([100] direction) and the c-axis ([001] direction) as 0.2–0.6 ${m}_{0}$ and 0.5–3.3 ${m}_{0}$ at low doping degrees, respectively. Taking TiO₂ with Nb-doping concentration of 2.08 at. % as a benchmark, the relative mobility of TiO₂ with other doping concentrations is calculated by equation (2), as displayed in table 3. The mobility reduces with the increasing Nb concentration, which is consistent with the previous experimental results [6, 37].

Table 3. Effective mass, electron concentration, mobility and conductivity of TiO₂ doped with various concentrations of Nb.

	Nb concentration (at.%)
	2.08	4.17	6.25	10.42	14.58
Effective mass in the [100] direction, ${m}_{a}^{* }$ (${m}_{0}$)	0.226	0.546	0.359	0.384	0.768
Effective mass in the [001] direction, ${m}_{c}^{* }$ (${m}_{0}$)	1.809	2.764	4.983	1.861	5.157
Average effective mass, ${m}_{n}^{* }$ (${m}_{0}$)	0.319	0.745	0.520	0.522	1.072
Electron concentration, ${n}_{0}$ (cm−3)	9.91 × 1020	3.65 × 1021	5.44 × 1021	8.59 × 1021	1.19 × 1022
Relative electron mobility, $\mu$	1	0.21	0.20	0.12	0.04
Relative conductivity, ${\rm{\sigma }}$	1	0.77	1.10	1.04	0.48

The electrons in Nb-doped TiO₂ follow Fermi–Dirac distribution $f(E),$ which is expressed by the following equation (5):

$$\begin{eqnarray}&&f(E)=\displaystyle \frac{1}{1+\exp \left(\displaystyle \frac{{E}_{i\,}+{E}_{F}}{{k}_{B}T}\right)}\end{eqnarray} \tag{ 5 }$$

Where ${E}_{i}\,$is the energy level at the base of the conduction band, ${E}_{F}$ is Fermi level, ${k}_{B}$ is Boltzmann constant, T is the thermodynamic temperature. The electron concentration ${n}_{0}$ in the conduction band can be calculated by equation (6).

$$\begin{eqnarray}&&{n}_{0}=\displaystyle \frac{1}{V}\displaystyle \int f(E){G}_{c}(E)dE\end{eqnarray} \tag{ 6 }$$

Where V is the volume, ${G}_{c}(E)$ is the DOS near the Fermi level. In addition, the conductivity ${\rm{\sigma }}$ is a function of the electron concentration ${n}_{0}$ and the electron mobility $\mu :$

$$\begin{eqnarray}&&\sigma ={n}_{0}q\mu \end{eqnarray} \tag{ 7 }$$

where q = 1.6 × 10⁻¹⁹ C is the elementary charge. According to equations (2)–(7), The electron concentration and the relative conductivity of Nb-doped TiO₂ are both calculated, as shown in table 3. The calculated electron concentration value of 9.91 × 10²⁰ cm⁻³ is in good agreement with experimentalvalue [44], and gradually increases to 1.19 × 10²² cm⁻³ with the increase of Nb-doping concentration. In the case of low Nb-doping (<6.25 at.%), the conductivity of Nb-doped TiO₂ reduces slightly with the increase of Nb-doping concentration, which is also consistent with the experimental results [6]. When the doping concentration is 6.25 at.%, the conductivity will reach the maximum value and then slowly decreases. When the doping concentration reaches 14.58 at.%, the conductivity will rapidly decrease. The variation trend of conductivity at high Nb-doping accords well with the experimental results [37].

In order to understand the optical properties of TNO, the absorption spectra and reflectance spectra for the pure TiO₂ and Nb-doped TiO₂ were calculated respectively, as shown in figure 7. In the figure 7(a), the absorption edge of the pure TiO₂ is near 3.2 eV, which shows that TiO₂ is a direct band gap semiconductor. The absorption edge gradually shifts to the higher energy side as the Nb concentration increases, and appears blue-shift phenomenon. This is consistent with the results of previous experimental studies [45, 46]. The absorption spectra of Nb doping concentration in the range of 2.08 at.% to 10.42 at.% show an absorption peak near 0.7eV. When Nb concentration reaches 14.58 at.%, the absorption peak moves to the high energy level. The absorption intensity in the visible region increases. This is probably due to free electrons in the Ti 3d and Nb 4d orbitals in the conduction band. In the previous experiment, Kurita et al [47] prepared TiO₂ films with different Nb-doping concentrations by PLD technique, and their absorption spectra showed similar results.

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It can be seen from figure 7(b) that the reflectivity of doped TiO₂ in visible region is lower than that of pure TiO₂, but in infrared region, the reflectivity of doped TiO₂ is much higher than that of pure TiO₂. The reflectivity of doped TiO₂ is below 10% in the visible region, which is better than that in infrared region and ultraviolet region, and the absorption coefficient is relatively low, which indicates that doped TiO₂ possesses a relatively high transmittance in the visible region. However, when the concentration exceeds 14.58 at.%, the transmittance will deteriorate because of the larger absorption coefficient in the visible region.