^{1}

^{2}

^{1}

^{3}

^{3}

^{3}

^{1}

^{1}

^{3}

^{1}

^{2}

^{3}

The _{2} and their impact on the transport coefficients are reported. As the doping of the Zr or Ti interstitials in the TiO_{2}, the lattice Ti^{4+} ions acquire the excess electrons so reduced to the Ti^{3+} or Ti^{2+} ions. However, the Cu interstitials could not lose enough electrons to reduce the lattice Ti^{4+} ions. Furthermore, the Ti or Cu interstitials in the ZrO_{2} also are unable to promote the lattice Zr^{4+} ions to form the lattice Zr^{3+} or Zr^{2+} ions. The high transport coefficients are observed in the defected TiO_{2} with the Ti or Zr interstitials as the high concentration of the Ti^{3+} or Ti^{2+} ions. So, the Zr interstitials are the favorable choice for the extra-doping to improve the transport properties in the TiO_{2}-based resistive random access memory.

In the recent years, the resistive random access memory (ReRAM) has been extensively studied due to its high operation speed, the long retention time, and the low power consumption. It always uses the metal-insulator-metal as the basic structure. Various transition metal oxides, such as TiO_{2}, CuO, HfO_{2}, and ZrO_{2}, are applied as the insulator of the ReRAM cells [_{2} is one of the most promising materials for the insulator in ReRAM. Up to now, one of the major challenges is the large variation of the switching parameters induced by the random formation of the conduction path under the external electrical fields. It needs to clarify the resistive switching mechanisms to solve this problem and finally improve the reliability of these devices. For the insulator of the rutile TiO_{2}, the leakage currents are often explained with the gap states originating from the point defects, such as the oxygen vacancies and Ti interstitials [_{2}, it is necessary to investigate the role of the interstitials in the conduction path for the resistive switching, such as Cu interstitials and Zr interstitials. Many novel systems, such as Pt/ZrO_{2}/TiO_{2}/Pt, Pt/Ti/TiO_{2}/Pt, Cu/ZrO_{2}:Cu/Pt, and Cu/ZrO_{2}:Ti/Pt, have been proposed to explain the resistive switching mechanism [_{2}.

The structural model of the perfect rutile TiO_{2} is presented with a primitive cell (lattice parameters _{2} at the same coordinates as Ti interstitials as shown in Figure _{2} (_{2} with the Cu substitutions or the Zr substitutions for the Ti atoms at (2.535, 2.535, 1.480), (2.535, 2.535, 7.398), and (2.535, 2.535, 13.316) as shown in Figure

Configuration of the defected TiO_{2} with the interstitials (a), the defected ZrO_{2} with the interstitials (b), and the defected TiO_{2} with the substitutions (c).

We explore the DMol^{3} program to carry out the spin polarized density functional calculations. These geometries are optimized using the double-numeric quality basis set (DNP) equal to 3.5, together with the PBE (Perdew, Burke, and Ernzerhof) gradient-corrected functional to describe the exchange and correlation effects [^{−5} Ha, 2 × 10^{−3} Ha/Å, and 5 × 10^{−3} Å, respectively. We further use Virtual NanoLab program to calculate the transmission coefficient with the DFT-PBE function at 300 K. The cutoff of the grid mesh is set to 40 hartree. The basis sets of double zetas and polarization orbitals (DZP) are performed in the transport simulations.

Figure _{2} with the Cu interstitials (a), Ti interstitials (b), Zr interstitials (c), and the corresponding white-black views (d–f). The gray balls and the red balls indicate Ti-atoms and O-atoms, respectively. The brown balls, the blue balls, and the green balls separately stand for the Cu interstitials, Ti interstitials, and Zr interstitials. We set the spectrum of the deformation electron density to the blue-green-red from −0.1 to 0.1 electrons/Å^{3}. The deficiencies for the electrons are indicated in blue color, while the enrichment is in red color. In Figure

Deformation electron density in _{2} with the Cu interstitials (a), Ti interstitials (b), Zr interstitials (c), and the corresponding white-black views (d–f).

Figure _{2} with the interstitials. Ti(1)–Ti(6) indicate the lattice Ti-ions, and i(7)–i(9) mean the interstitials with Number 7–Number 9 as shown in Figure _{12}Cu_{3}O_{24}, Ti_{15}O_{24}, and Ti_{12}Zr_{3}O_{24} to stand for the structures in Figure _{2} are 1.721 eV. The Mulliken charges of the lattice Ti-ions in Ti_{12}Zr_{3}O_{24} (49.9% of 1.721 eV) are smaller than those in Ti_{15}O_{24} (66.8%) and Ti_{12}Cu_{3}O_{24} (80.4%). The Mulliken charges for the Zr interstitials in Ti_{12}Zr_{3}O_{24} are larger than those for the Ti interstitials in Ti_{15}O_{24} and Cu interstitials in Ti_{12}Cu_{3}O_{24}. The Mulliken charges for the lattice Zr ions in the perfect ZrO_{2} are 2.292 eV. So the Zr interstitials in Ti_{12}Zr_{3}O_{24} lose more electrons and the corresponding Ti atoms are reduced to the Ti^{3+} or Ti^{2+} ions with the smallest Mulliken charges. The electric field of 10.4 MV/cm redisperses the electrons which transfer between the lattice Ti atoms and the interstitials. It would explain the difference of Mulliken charges among the Ti(1)–Ti(6) atoms.

Mulliken charges in the defected TiO_{2} with the interstitials.

Figure _{12}Cu_{3}O_{24} are shorter than those in Ti_{15}O_{24} and Ti_{12}Zr_{3}O_{24}.

Distances among the lattice Ti-ions and the interstitials in the defected TiO_{2} with the interstitials.

Figure _{2} with the Cu interstitials (a), Ti interstitials (b), and Zr interstitials (c). The curves with the light green color, the red color, the blue color, the dark green color, and the pink color separately indicate the p states, the d states, the sum states, the DOS induced by the Ti-ions (Ti(1)–Ti(6)), and the DOS induced by the interstitials. In Figure ^{3+} or Ti^{2+} trap centers.

Partial density of states for the defected TiO_{2} with the Cu interstitials (a), Ti interstitials (b), and Zr interstitials (c).

In Figures _{2} can induce the lattice Ti atoms to Ti-ions with the lower valence. Next, we continue to consider the defected ZrO_{2} with the Ti interstitials and the Cu interstitials.

Figure _{2} with the Ti interstitials (a, c) or Cu interstitials (b, d). Figure _{2}. In Figure

Deformation electron density in _{2} with the Ti interstitials (a) and Cu interstitials (b); partial density of states for the defected ZrO_{2} with the Ti interstitials (c) and Cu interstitials (d); (e) density of states for the perfect ZrO_{2}.

In Figure _{2} in Figure

We have addressed the defected TiO_{2} and the defected ZrO_{2} with the interstitials. The Zr interstitials in the defected TiO_{2} reduce the lattice Ti-ions to the Ti^{3+} or Ti^{2+} ions. However, the Ti interstitials in the defected ZrO_{2} do not reduce the lattice Zr-ions. Next, the Cu substitutions or the Zr substitutions for the lattice Ti atoms in the defected TiO_{2} would be considered.

Figure _{2} with Cu substitutions (a) and Zr substitutions (b). The gray balls, the red ball, the pink balls, and the green ball indicate the Ti atoms, the O atoms, the Cu substitutions, and the Zr substitutions, respectively. The data with green color means the Mulliken charges. The Mulliken charges of the Cu substitutions and Zr substitutions in the defected TiO_{2} are larger than those in the perfect CuO (0.735 eV) and in the perfect ZrO_{2} (2.292 eV). The Mulliken charges of the Ti-ions are larger than 1.59 eV, which appear to be distinct to those in Figure _{2} always induce the formation of the Ti^{4+} ions.

Partial sectional view in _{2} with the Cu substitutions (a) and Zr substitutions (b).

Figure _{2} with Cu substitutions (a) and Zr substitutions (b). The curves with the light green color, the red color, the blue color, and the dark green color separately indicate the p states, the d states, the sum states, and the DOS induced by the substitutions. The Fermi energy levels locate on the top of the valence band maximum but with the lower energy in Figure _{2} with the interstitials.

Partial density of states for the defected TiO_{2} with the Cu substitutions (a) and Zr substitutions (b).

Figure _{2} with the Cu interstitials (a), the Ti interstitials (b), the defected TiO_{2} with the Zr interstitials (c), the Ti interstitials (d), and the Cu interstitials (e). In Figures _{2} with the interstitials, while a transmission gap of 0.8 eV in Figure _{2} with the Ti interstitials was found. In Figure _{2} with the Ti interstitials. In contrast, the large blocks of the electrons deficiency in Figure _{2} with the Ti interstitials. The relative Fermi energy levels also could explain these phenomena. The strong hybridization between O-2p and Ti-3d orbitals decreases the transport coefficients from −1.5 eV to −0.5 eV in Figure _{2} with the Zr interstitials. So the advanced transport coefficients are found in the defected TiO_{2} with the Ti interstitials and then in the defected TiO_{2} with the Zr interstitials and the worst in the defected TiO_{2} with the Cu interstitials.

Transmission coefficients for the defected ZrO_{2} with the Cu interstitials (a), Ti interstitials (b), the defected TiO_{2} with the Zr interstitials (c), Ti interstitials (d), and Cu interstitials (e).

The formation energies of the Ti interstitials, the Zr interstitials, and the Cu interstitials are 5.95 eV, 8.86 eV, and 10.24 eV. The better stability of the Ti interstitials self-doping in the TiO_{2} improves the transport properties. For the Cu atoms or Zr atoms, their steady coordination valences with O atoms are +2 in CuO or +4 in ZrO_{2}. In the above discussions, the doping of the Zr interstitials in TiO_{2} reduces the lattice Ti^{4+} ions to the Ti^{3+} or Ti^{2+} ions and leads to the higher transport coefficients than the Cu interstitials do. So, for the metal atoms with the valences of +4 or more in their binary oxides, their interstitials doping in TiO_{2} could further reduce the Ti^{4+} ions to the Ti^{3+} or Ti^{2+} ions rather than those with the lower valences in the binary oxides, such as +3 or +2. We use the Fe interstitials (+3) and Hf interstitials (+4) to verify the conclusion [_{12}Hf_{3}O_{24} are much smaller than those in Ti_{12}Fe_{3}O_{24}.

Mulliken charges in the defected TiO_{2} with the Fe or Hf interstitials.

We focus on_{2} and their impact on the transport coefficients. We find that the Zr or Ti interstitials in the TiO_{2} lose the electrons and induce the formation of the Ti^{3+} ions with the low Mulliken charges. The Cu interstitials also reduce the lattice Ti^{4+} ion but with less extent. By contrast, the Ti or Cu interstitials in the ZrO_{2} do not induce the formation of the lattice Zr^{3+} ions. Furthermore, the Cu or Zr substitutions for the Ti atoms in the TiO_{2} also keep the lattice Ti-ions in the +4 valence. As the formation of the Ti^{3+} trap centers, the Ti or Zr interstitials in TiO_{2} lead to the higher transport coefficients than the Cu interstitials do. Finally, we propose that the doping of the metal atoms with the valences of +4 or more in their oxides may be beneficial for the formation of the Ti^{3+} ions in TiO_{2} and so improves the transport properties. This work may be helpful to guide the doping principle in the metal/TiO_{2}/metal structure of the resistive random access memory [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors acknowledge the support from the National Natural Science Foundation of China under Grants nos. 61076102 and 61272105 and Natural Science Foundation of Jiangsu Province of China under Grants nos. BK2012614 and BK20141196.

_{2}resistive switching memory

_{2}(110) surface

_{2}crystals

_{2}

_{2}

_{2}(110): Surface rearrangement and reactivity studied using elevated temperature scanning tunneling microscopy

_{2}layer formation in TiO

_{2}-based forming-free resistive random access memory

_{2}

_{2}calculated with periodic and embedded cluster density functional theory

_{x}Layer with CF

_{4}plasma treatment