A DFT + U study on structural, electronic, vibrational and thermodynamic properties of TiO₂ polymorphs and hydrogen titanate: tuning the Hubbard 'U-term'

As it was said above, anatase, rutile, TiO₂–B and hydrogen titanate were relaxed in order to find their optimum geometries. In table 1 the lattice parameters a, c and c/a ratio at different U values are listed.

The incorporation of U term affects the lattice parameters, expanding the cell while the U value increases. That is the reason why it is needed to find the right U value, which balances the accuracy of the predicted geometrical structure and electronic properties for each system. In table 2 it can be seen that Ti–O bond lengths in every system increase when U takes higher values. However, the changes are not significant or drastic.

In table 3 a comparison of the band gap widths at different U term values and experimental results obtained from literature are summarized [33–35]. As the U term value increases, the band gap increases as well.

The density of states is plotted in figure 2, each system at promising optimum U term value, namely, U = 4 eV for anatase and TiO₂–B, U = 5 eV for rutile and hydrogen titanate. Band gap energies are in good agreement with those found in previously cited literature for TiO₂–B and hydrogen titanate. However for the cases of anatase and rutile a selection of higher U values leads to a major structural deviation, that is why we have selected values of U = 4 eV for anatase and U = 5 eV for rutile (see table 4). Despite the band gap energies are far from experimental reported data an improvement can be seen when U term in taken into account in the calculation.

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In these curves a significant contribution of O 2p states to the valence band, and an important contribution of Ti 3d states in the conduction band, can be seen. This is a well-known feature for bulk TiO₂ polymorphs.

In order to determine the appropriate values of U for each system, we have compared the calculated band gap widths for a set of U values (3, 4 and 5 eV) with experimental band gap values. However, it is not only necessary to fit the band gap width but also it is needed to maintain the experimental lattice parameters. This information can be seen in table 4, for TiO₂–B polymorph and hydrogen titanate there is no structural experimental information available and for this reason the values calculated for different U parameters were compared with the results calculated using U = 0 eV.

In order to obtain the zone-center phonons of the TiO₂ polymorphs and hydrogen titanate, the dynamical matrix should be diagonalized for q = 0. Through factor analysis [36], in the I4₁amd, P4₂/mnm, C12/m1 and C2/m space groups for anatase, rutile, TiO₂–B and hydrogen titanate bulks respectively, the irreducible representations of the optical vibrations are as follows:

$$\begin{eqnarray*}&&\begin{array}{l}{\rm{anatase}}:{{\rm{\Gamma }}}_{{\rm{opt}}}={A}{1}_{{\rm{g}}}+{A}{2}_{{\rm{u}}}+2{B}{1}_{{\rm{g}}}+{B}{2}_{{\rm{u}}}+2{{E}}_{{\rm{g}}}+2{{E}}_{{\rm{u}}},\\ {\rm{rutile}}:{{\rm{\Gamma }}}_{{\rm{opt}}}={A}{1}_{{\rm{g}}}+{A}{2}_{{\rm{g}}}+{A}{2}_{{\rm{u}}}+2{B}{1}_{{\rm{u}}}+{B}{1}_{{\rm{g}}}+{B}{2}_{{\rm{g}}}+{{E}}_{{\rm{g}}}+3{{E}}_{{\rm{u}}},\\ {{\rm{TiO}}}_{2}\mbox{--}{B}:{{\rm{\Gamma }}}_{{\rm{opt}}}=12{{A}}_{{\rm{g}}}+5{{A}}_{{\rm{u}}}+6{{B}}_{{\rm{g}}}+10{{B}}_{{\rm{u}}},\\ {\rm{hydrogen}}\,{\rm{titanate}}:{{\rm{\Gamma }}}_{{\rm{opt}}}=11{{B}}_{{\rm{g}}}+11{{A}}_{{\rm{u}}}+22{{B}}_{{\rm{u}}}+24{{A}}_{{\rm{g}}}\end{array}.\end{eqnarray*}$$

The modes with subscript g (gerade) are Raman-active and those with subscript u (ungerade) are infrared-active, while the modes represented with the letter E are degenerate.

In all the cases, the phonons density of states have been obtained, and Ti, O and H atoms are projected in black, red and blue curves, respectively, as it can be seen in figure 3. In TiO₂ polymorphs the low values of frequency correspond mainly to titanium atoms, and high values to oxygen atoms, anatase and rutile plots agree with those from literature [37]. In the case of hydrogen titanate we can see a wide gap among curves, there are two picks at high energy values which correspond to the light hydrogen atoms.

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In figure 4 the histograms of each bulk system are presented, the bars represent the group of vibration modes in a specific range of frequency. The vertical black lines are the vibration modes calculated at DFT U = 0 eV level for comparison. From these charts, it can be noticed that the implementation of the U term has an effect on the vibrational properties, developing different vibration modes at different frequencies. How this influences the thermodynamic properties will be analyzed in the following section.

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For bulk rutile, there are lots of data reported: in table 5 a comparison of this information and our results is shown [38–42]. Here it can be seen clearly how the Hubbard implementation changes the vibrational properties, frequency values of vibrational modes differ and shift significantly when different values of U term are taken into account in the calculation. Reported theoretical results without U term approach better to experimental information.

Once the phonon frequencies set is obtained, thermodynamic properties of the systems can be calculated by quasi harmonic approximation. According to the following expressions, entropy, vibration energy and Helmholtz free energy can be determined:

$$\begin{eqnarray*}F & = & E-TS,\\ E & = & \displaystyle \sum _{q\nu }\hslash \omega (q\nu )\left[\displaystyle \frac{1}{2}+\displaystyle \frac{1}{{e}^{\tfrac{\hslash \omega (q\nu )}{{k}_{B}T}}-1}\right],\\ S & = & -\displaystyle \frac{\partial F}{\partial T}=\displaystyle \frac{1}{2T}\displaystyle \sum _{q\nu }\hslash \omega (q\nu )\coth ({e}^{\tfrac{\hslash \omega (q\nu )}{2{k}_{B}T}})-{k}_{B}\displaystyle \sum _{q\nu }\mathrm{ln}\left[2\,\sinh \left(\displaystyle \frac{\hslash \omega (q\nu )}{2{k}_{B}T}\right)\right].\end{eqnarray*}$$

Also, specific heat at constant volume can be calculated as follows:

$$\begin{eqnarray*}&&{C}_{{\rm{V}}}={\left(\displaystyle \frac{\partial E}{\partial T}\right)}_{V}=\displaystyle \sum _{q\nu }{k}_{B}{\left(\displaystyle \frac{\hslash \omega (q\nu )}{{k}_{B}T}\right)}^{2}\left\{\displaystyle \frac{{e}^{\tfrac{\hslash \omega (q\nu )}{{k}_{B}T}}}{{({e}^{\tfrac{\hslash \omega (q\nu )}{{k}_{B}T}}-1)}^{2}}\right\}.\end{eqnarray*}$$

Properties of solids can be extrapolated at temperatures higher than 0 K by vibration harmonic approximations. Based on these quasi-harmonic approximations, we have obtained the phonon frequencies at five different volumes, namely, 3% and 1.5% reduced volume, 0%, and 3% and 1.5% expanded volume. In this manner, we have calculated a set of thermodynamic parameters for TiO₂ polymorphs and hydrogen titanate bulks. In figures 5–8 we have plotted enthalpy and entropy versus temperature for the four systems as well as C_p (specific heat at constant pressure) and C_v (specific heat at constant volume) versus temperature, respectively. In black crosses experimental data for anatase and rutile from literature was added [43].

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It can be seen in these four figures that the three TiO₂ polymorphs behave in a similar way, while hydrogen titanate takes higher values in every thermodynamic property studied here. In the enthalpy versus temperature plot (figure 5), we can see that up to 200 K the three TiO₂ polymorphs take very similar values at standard DFT, however at DFT + U level the enthalpy values are overestimated. In the case of entropy (figure 6) the curves of each system can now be distinguished, nevertheless the curves fit better with experimental information when rutile takes a U = 5 eV value and anatase U = 4 eV than DFT standard calculation. In C_v (and C_p) versus temperature plot (figures 7 and 8), from 400 K up once again, the three TiO₂ polymorphs behave similarly, approaching an asymptotic behavior at higher temperatures. Differences in these three TiO₂ polymorphs are not significant.

Thermal volumetric expansion can be determined and are plotted as shown in figure 9. TiO₂ polymorphs have the same behavior; hydrogen titanate does not experience any significant volumetric change when temperature is increased. Instead, a volume contraction is observed near 200 K which is correlated to the presence of several imaginary vibrational modes, that only appear for the cell volumes associated to that temperature, but that does not appear for the equilibrium values associated at 0 K. This fact could be connected to a possible phase transformation from hydrogen titanate to TiO₂(B) in the same range of temperatures [44]. Anatase and rutile lead to bigger volumetric variations.

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Thermodynamic properties fluctuate significantly when the Hubbard term is considered in our analysis. The previous differences noticed for phonon frequencies have an effect on the results of thermodynamic properties calculation.

In table 6 bulk modulus and volume variation using DFT and DFT + U at 0 K and 298 K is listed to compare the effect of addition of Hubbard parameter in our calculations [13, 45, 46].

The addition of Hubbard parameter leads to an increase in cell volume being less significant for TiO₂–B and hydrogen titanate. The bulk modulus varies when U term is considered, being larger for anatase and TiO₂–B, and smaller for rutile and hydrogen titanate. In the cases of TiO₂–B and hydrogen titanate few data in literature is found, in order to compare our results, we have added two rows called 'DFT U = 0' which correspond to our results at a DFT standard level.