Electronic and optical properties of metal-doped TiO$_2$ nanotubes: Spintronic and photocatalytic applications

Due to their characteristic geometry, TiO$_2$ nanotubes (TNTs), suitably doped by metal-substitution to enhance their photocatalytic properties, have a high potential for applications such as clean fuel production. In this context, we present a detailed investigation of the magnetic, electronic, and optical properties of transition-metal doped TNTs, based on hybrid density functional theory. In particular, we focus on the $3d$, the $4d$, as well as selected $5d$ transition-metal doped TNTs. Thereby, we are able to explain the enhanced optical activity and photocatalytic sensitivity observed in various experiments. We find, for example, that Cr- and W-doped TNTs can be employed for applications like water splitting and carbon dioxide reduction, and for spintronic devices. The best candidate for water splitting is Fe-doped TNT, in agreement with experimental observations. In addition, our findings provide valuable hints for future experimental studies of the ferromagnetic/spintronic behavior of metal-doped titania nanotubes.

In particular, a titania nanotube (TNT) is an effective nano-photocatalyst that directly splits water [18,19], and degrades environmental pollutants [8] under sunlight. Furthermore, it is used in solar energy conversion [20] due to the good locations of its conduction band (CB) and valence band (VB) edges with respect to hydrogen formation and oxidation energy [21]. Moreover, the highly ordered nanotube geometry and large internal surface area are very useful as a unidirectional electric channel for the photogenerated electrons [22]. However, the bandgap of TNT (3.18-3.23 eV [23,24]) restricts its applications in photocatalytic processes because of the limited absorption in the visible-light range. Therefore, engineering the bandgap of TNT by dopants to increase its photosensitivity to visible light is a major target in photocatalyst studies. On the other hand, Co- [25] and Ni-doped [26] TNTs can be used as dilute magnetic semiconductors.
Most publications in the field have been concerned with mechanisms that decrease the bandgap of TNTs, and shift the absorption edges towards the visible-light range. Doping is a common method for tuning the bandgap of semiconductors. Experimentally, a large number of doped TNTs was prepared, e.g., Refs. [27][28][29][30][31]; and theoretical studies, based on density functional theory (DFT), include, e.g., Refs. [32][33][34].
In view of the relative simplicity of preparing metal-doped TNTs [37] (M-TNTs), further theoretical studies are highly needed to elucidate systematically the photocatalytic and magnetic properties of such systems. This paper is devoted to such a task, in particular, we 2 present results of a systematic and accurate-based on hybrid density funcional theoryinvestigation of the structural, electronic, and optical properties of 3d, 4d, and selected 5d metal-doped TNTs, in order to contribute to a better understanding of available experimental results for photocatalytic and spintronic properties, as well as for providing hints for future experiments.
With respect to spintronics application, magnetic sensors and non-volatile magnetic memories are conceivable. As compared to metal-based spintronics, metal-oxide structures are more versatile because of the ability to control the potential variation and spin polarization by external voltages [38]. Several spintronic experiments have been performed for carbon nanotubes using a two-terminal spin valve geometry [39,40]. This experimental setup makes it difficult, however, to separate spin transport from other effects, such as Hall effect, anisotropic magnetoresistance [41], magneto-Coulomb [42] and interference effects [43]. A four-terminal non-local spin valve setup [41,44,45] with a single-wall nanotube can separate the spin current from the charge current [46]. Also, there are promising results for one-dimensional perovskite spintronic devices at a temperature lower than room temperature [47] (which is a key issue for commercial applications). Difficulties in device fabrication include the interface spin transport, and the positioning of nanowires or nanotubes with respect to the other components. The development of the system architecture may help overcome the difficulties associated with traditional electronics, and allow to develop onedimensional spintronics technologies with good scalability, and lower power dissipation [38].
The details of the calculational approach are given in Section II. In Section III, we address the structural and magnetic properties. The main part of this study, the electronic structures of M-TNTs, is presented in Section IV, followed by the optical and photocatalytic properties in Section V. A brief summary is given in Section VI.

II. COMPUTATIONAL DETAILS
All the calculations were carried out using plane-wave pseudopotentials in the Vienna ab initio simulation package [48]. Spin-polarized calculations were employed to determine the structural, electronic, and magnetic properties. The generalized gradient approximation in the scheme of Perdew-Burke-Ernzerhof was used as an exchange-correlation functional for structure optimization. The plane-wave functions with cutoff energy 550 eV and 1 × 1 × 3 k-mesh based on the Monkhorst-Pack method were utilized to obtain a converged force of 0.01 eV/Å, and the total tolerance energy is 10 −6 eV. A periodic supercell along the tube axis (z axis) was considered with a vacuum distance of 20Å between nanotubes in x and y directions to prevent the interaction between the neighboring TNTs. The convergence of the results was not affected when the parameters (cutoff energy, supercell size, and k-mesh) were increased. The Heyd-Scuseria-Ernzerhof hybrid functional (HSE) [49] was employed to calculate the formation energies, electronic structures, and optical properties.
The exchange-energy functional in the HSE scheme is written as a linear combination of short-range (SR) and long-range (LR) terms. The SR term is a PBE exchange functional (E PBE,SR X (µ)) mixed with a certain percentage of the exact exchange Hartree-Fock (E HF,SR X (µ)) contribution, while the LR term is defined by the PBE exchange functional (E PBE,LR X (µ)). The range-separation (screening) parameter µ is usually and also in this work chosen to be 0.2Å −1 . Therefore the exchange-correlation energy functional reads: where a is the called exchange mixing coefficient. The standard choice of a in the HSE06 package is 25%, but in order to reproduce the experimental TiO 2 bulk bandgap [50], one has to choose a slightly smaller value, 22%. However, both these a values strongly overestimate the TNT bandgap. In order to test the sensitivity of the bandgap with respect to variations of the mixing coefficient, we have changed a from 10% to 28%: the bandgap is found to be approximately given by 3.0 eV for 10%, and 4.3 eV for 28%, with an almost linear increase in between; cf. Table I. The best a value, reproducing the experimental TNT bandgap of 3.2 eV, is 14%, which we have chosen in the following. Our results for the density of states (not shown) also indicate that when increasing a the conduction band shifts as a whole to higher energy (relative to the Fermi energy), while the valence band stays rigid.
The dielectric function, ε(ω) = ε 1 (ω) + iε 2 (ω), describes the optical response at the angular frequency ω. First, ε 2 (ω) is calculated on the basis of the standard golden-rule expression, then ε 1 (ω) is found by employing the Kramers-Kronig relation. Finally, the absorption spectrum is determined by optimal mixing parameter for TNT as compared to bulk TiO 2 (a = 22%) means that the electrons in the nanotube are less localized than those in the bulk system, hence the TNT electrons are easily polarizable, implying good screening [51]. This also is an indication that there is no need for more sophisticated many-body technqiques like dynamical mean-field theory.

III. STABILITY OF M-DOPED TNTS
TiO 2 anatase nanotubes have been investigated experimentally and theoretically, see, e.g., [52,53] and [54,55], respectively, including details of the geometries and their stabilities [33,34,54,55].  Tables II, III, and IV. The stability of the metal-doped TNT (M-TNT) is determined from the formation energy (E f ): where E M−TNT and E TNT denote the total energies of metal-doped and pristine TNT, respectively; µ Ti and µ M are the chemical potentials of Ti and the dopant atom, the latter assumed to be given by the energy of the isolated metal atom. energies (E f (eV)), magnetic moments (µ B ), and bandgap (eV) of selected 3d-metal doped TNTs.
The pristine system (Ti column) is included for easy reference. The last three rows indicate whether the respective system is useful for spintronic, optical, and photocatalytic applications: the means "clear improvement compared to pristine TNT", and the "no improvement", w.r.t. that application.   (Tables II, III, and IV).
Regarding the magnetic moment, the difference between the number of outer shell electrons in the metal dopant and the Ti atom determines the magnetic properties of M-TNTs as is apparent, e.g., for Sc-to Fe-TNTs. For the nearly full outer shell Co, the coupling between the outer shell electrons can explain the magnetic moment of Co-TNT: two outer shell electrons are coupling and the third one is unpaired (low spin state), hence the net magnetic moment is 1 µ B to a good approximation.
On the other hand, for Co one needs one electron to fill the outer shell, thereby a hole will be created in Co-TNT, and the magnetic moment becomes also 1 µ B . For the full-filled outer shell atoms (Ni, Cu, and Zn), the magnetic moments are 0 for Ni due to the closed (inert) shell, 1 µ B for Cu (the oxidation number is (+3)) and a hole is created, and 2 µ B for Zn (the oxidation number is (+2)) and two holes are created; see Table II. The same trend appears in the 4d-TNTs (Table III) (Table IV).

IV. ELECTRONIC STRUCTURE
The density of states (DOS) and projected density of states (PDOS) for pristine TNT are shown in Figs. 2(a,b) (Fig. 2(b)).
In order to structure the presentation, we discuss the electronic structures according to the groups in the periodic table. For 3B group dopants (Sc, Y, La), due to their oxidation number of (+3) as compared to (+4) for Ti, the DOS is asymmetric between the two spin    (Figs. 3(g,h)).
Since the "scissors" operation is applied; see [34] and references therein.
observed that Nb-TNT is an n-type conductor [29], consistent with our results. Also, it was found experimentally that Nb [59] and Ta [60] dopings enhance the photocatalytic activity of TNT for water splitting. In addition, ferromagnetic behavior was reported for V-TNT [35]. Clearly, due to the metallic spin-up states of Nb-and Ta-TNTs at the Fermi energy, these systems may be useful for spintronic applications. Tc doped bulk TiO 2 was synthesized [67].
Regarding the 8B group, first two columns (Fe, Ru, Co, Rh), these atoms have nearly full outer shells. As compared to the previous dopants, the created states are more spread inside the bandgap, and appear near the edges of the VB and the CB as well, which implies a stronger reduction of the bandgap (Figs. 6(e,g) and Figs. 7(a,c)). The bandgaps are 1.7, 1.6, 1.5, and 1.2 eV for Fe-, Ru-, Co-, and Rh-TNTs, respectively. Except for the Rh dopant, all of them will be good candidates for enhancing the optical and photocatalytic activities of TNT. For Fe doping, an improvement of the photocatalytic activity was found experimentally [29,68]. Also Ru-TNT showed a reduced bandgap as compared to TNT in an experiment [31]. We note that ferromagnetic behavior was experimentally observed for Co-TNT [25].
We turn now to the closed outer shell atoms, namely the last column in the 8B group (Ni, Reduced bandgaps, compared to pristine TNT, have been observed experimentally for Cuand Ag-TNTs, and for Au doped TiO 2 nanoparticles [29,71,72]. This dopant group hence can only improve the optical activity. The last closed outer shell group is 2B (Zn, Cd). The effect of both dopants is the same in the DOS (Figs. 8(g,i)), with a small difference, however, in the location of the created midgap states. Figures 8(h,j)  its range increase (see, e.g., Fig. 9(b) as compared to Fig. 9(a)).
In Fig. 10(a)  After the arrival of the photogenerated charge pairs at the M-TNT surface, the photogenerated electrons will reduce the adsorbed H 2 O on M-TNT to form H 2 fuel gas, and the photogenerated holes will oxidize H 2 O to form O 2 on different active surface sites. Also, the photogenerated electrons can be used to reduce the adsorbed CO 2 on the M-TNTs to several natural fuels ( Fig. 10(a)). The two previous processes can only occur when the CBE is more negative than the H + (protons which were produced from the water oxidation process) or CO 2 reduction potentials, and the VBE is more positive than the H 2 O oxidation potential.  eV/NHE), and the CBE has to be lower (more negative) than the redox potential of H + /H 2 (0 eV/NHE). Therefore, the bandgap of the photocatalyst has to be larger than 1.23 eV (∼ 1000 nm) to split water into H 2 and O 2 , which is the minimum Gibbs free energy for this process. Here, the band edges are measured with respect to the normal hydrogen electrode (NHE), and their determination is discussed in detail in many publications, see, e.g., Refs. [33,34] and references therein. For the CO 2 reduction, the CBE has to be lower (more negative) than the redox potential of the natural fuel/CO 2 . The positions of the band edges (which depend on the bandgap) are the main criterion for specifying a good photocatalyst for H 2 O splitting or CO 2 reduction. Figure 10(b) demonstrates that Fe-TNT is best for producing hydrogen (from water 20 splitting), methane, and methanol (from CO 2 reduction). The Co-and Ru-TNTs have the lowest bandgaps, but they are useful photocatalysts for generating methane fuel only. The Cr-TNT is the best candidate for producing all the considered clean fuels in this study. The other M-TNTs in Fig. 10(b) show photocatalytic activities which enables them to split H 2 O and reduce CO 2 reduction better than pristine TNT.

VI. SUMMARY
In this work we systematically discussed the electronic, magnetic, and optical properties of titania nanotubes doped with 3d, 4d, and selected 5d transition metals, in order to elucidate their potential for various applications. Our study has been based on hybrid density functional theory, which is known for leading to most accurate (in comparison to other  Tables II, III,  TNTs are expected to be preferential for photocatalytic applications (water splitting and carbon dioxide reduction) as compared to pristine TNT. Fe-doped TNT is the best candidate for water splitting and for the production of hydrogen, methane, and methanol fuels, while the Cr-and W-doped TNTs are best for water splitting and CO 2 reduction, i.e., for the production of clean fuels and, at the same time, for helping to decrease the CO 2 pollution.
However, in several cases (14 out of 24) the created midgap states prevent an enhancement of photocatalytic sensitivity. Our results compare favorably with available experimental 21 observations.

ACKNOWLEDGMENTS
Financial support from the Deutsche Forschungsgemeinschaft (project number 107745057, TRR 80) is gratefully acknowledged.