# Stability and Electronic Properties of TiO$_{2}$ Nanostructures With and Without B and N Doping 

D. J. Mowbray, ∗ J. I. Martinez, J. M. Garc´ıa-Lastra, K. S. Thygesen, and K. W. Jacobsen 

Center for Atomic-scale Materials Design, Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark 

(Received January 8, 2009; Published June 19, 2009) 

We address one of the main challenges to TiO$_{2}$-photocatalysis, namely band gap narrowing, by combining nanostructural changes with doping. With this aim we compare TiO$_{2}$’s electronic properties for small 0D clus- ters, 1D nanorods and nanotubes, 2D layers, and 3D surface and bulk phases using different approximations within density functional theory and GW calculations. In particular, we propose very small ( R ≲ 5 ˚ A) but surprisingly stable nanotubes with promising properties. The nanotubes are initially formed from TiO$_{2}$ layers with the PtO$_{2}$ structure, with the smallest (2,2) nanotube relaxing to a rutile nanorod structure. We find that quantum confinement effects – as expected – generally lead to a widening of the energy gap. However, substi- tutional doping with boron or nitrogen is found to give rise to (meta-)stable structures and the introduction of dopant and mid-gap states which effectively reduce the band gap. Boron is seen to always give rise to n -type doping while depending on the local bonding geometry, nitrogen may give rise to n -type or p -type doping. For undercoordinated TiO$_{2}$ surface structures found in clusters, nanorods, nanotubes, layers and surfaces nitrogen gives rise to acceptor states while for larger clusters and bulk structures donor states are introduced. 

Motivated by the world’s ever increasing need for cleaner burning fuels and more viable forms of renewable energy, hy- drogen production via photocatalysis has been intensely re- searched as a possible candidate for addressing these issues. Since the first experimental formation of hydrogen by pho- tocatalysis in the early 1980s [1], TiO$_{2}$ has been the catalyst of choice. Reasons for this include the position of TiO$_{2}$’s con- duction band above the energy of hydrogen formation, the rel- atively long lifetime of excited electrons which allows them to reach the surface from the bulk, TiO$_{2}$’s high corrosion resis- tance compared to other metal oxides, and its relatively low cost [2–4]. 

However, the large band gap of bulk TiO$_{2}$ ( ≈ 3 eV) means that only high energy UV light may excite its electrons. This effectively blocks most of the photons which pierce the atmo- sphere, typically in the visible range, from participating in any bulk TiO$_{2}$ based photocatalytic reaction. On the other hand, the difference in energy between excited electrons and holes, i.e. the band gap, must be large enough ( ≳ 1.23 eV) to dis- sociate water into hydrogen and oxygen. For these reasons it is of great interest to adjust the band gap ε$_{gap}$ of TiO$_{2}$ into the range 1.23 ≲ ε$_{gap}$ ≲ 2.5 eV, while maintaining the useful properties mentioned above [5]. 

With this aim, much research has been done on the influ- ence of TiO$_{2}$ nanostructure [6–10] and dopants [5, 11–17] on photocatalytic activity. For low dimensional nanostructured materials, electrons and holes have to travel shorter distances to reach the surface, allowing for a shorter quasi-particle life- time. However, due to quantum confinement effects, lower di- mensional TiO$_{2}$ nanostructures tend to have larger band gaps [18]. On the other hand, although doping may introduce mid- gap states, recent experimental studies have shown that boron and nitrogen doping of bulk TiO$_{2}$ yields band gaps smaller 

## I. INTRODUCTION 

than the threshold for water splitting [11, 12]. This suggests that low dimensional structures with band gaps larger than about 3.0 eV may be a better starting point for doping. 

The experimental synthesis and characterization of nanos- tructured materials is in general a costly and difficult task. On the other hand, modern electronic structure modelling has reached a level where large-scale calculations can provide re- alistic descriptions of structure and electronic properties. 

Based on our investigation, we suggest as a promising can- didate small ( R ≲ 5 ˚ A) TiO$_{2}$ nanotubes, with a hexagonal ABC PtO$_{2}$ structure (HexABC), which we find to be surpris- ingly stable, even in the boron and nitrogen doped forms. This stability may be attributed to their structural similarity to bulk rutile TiO$_{2}$, with the smallest nanotube having the same struc- ture as a rutile nanorod. 

A further difficulty for any photocatalytic system is control- ling how electrons and holes travel through the system [19]. For this reason, methods for reliably producing both n -type and p -type TiO$_{2}$ semiconducting materials are highly desir- able. So far, doped TiO$_{2}$ tends to yield only n -type semicon- ductors. We propose that p -type TiO$_{2}$ semiconducting mate- rials may be obtained by nitrogen doping surface sites of low dimensional materials. 

In this study we report the results of density functional theory (DFT) and GW calculations of the energetic stability and electronic structure of recently suggested (TiO$_{2}$)$_{n}$ clus- ters ( n ≤ 9) [6] and novel ( n , n ) TiO$_{2}$ nanotubes ( n ≤ 4) in the undoped, boron doped, and nitrogen doped forms. The formation energy E$_{form}$ , density of states (DOS), and energy gap ε$_{gap}$ , for these systems are compared with that for 2D HexABC and anatase layers, and 3D TiO$_{2}$ anatase surface, anatase bulk, and rutile bulk phases. Furthermore, we analyze how boron and nitrogen doping influences the DOS for these systems, and how their nanostructure may determine whether the resulting semiconductor is n -type or p -type. 

In Sec. II we describe the DFT and GW methodologies used to obtain the energies and electronic structure of the studied systems. We compare the energetic stability of the systems in Sec. III. In Sec. IV we discuss the electronic structures of 

