Relevance of non-equilibrium defect generation processes to resistive switching in TiO₂ Open Access

The resistive switching effect in metal oxide films is currently the focus of intense research owing to its potential as a low-power, high-speed, high-endurance, and non-volatile memory technology (ReRAM).^1–5 First documented in the 1960s,^6–9 the effect involves reversible switching between high and low resistance states by application of voltage pulses and has been observed in many oxide materials, including TiO₂, HfO₂, SrTiO₃, and Ta₂O₅.^10–15 Recent years have seen significant progress in understanding resistive switching; however, uncertainty still remains over the microscopic mechanisms responsible for the effect. There is a consensus that for many oxide materials the low resistance state involves an oxygen deficient conductive filament that bridges the electrodes.^4,16–18 In most cases, as grown devices are subjected to a forming step (in which a voltage stress is applied) to establish this filament.¹¹ Switching to the high resistance state then involves partial reoxidation of the filament.^4,19 One key issue often debated is whether defect generation under electrical bias, i.e., the rupturing of bonds leading to the creation of vacancy defects in the lattice, is important (particularly for forming). The generation of such defects, assisted by a non-equilibrium electric field and charge injection, is often discussed in the literature and included explicitly in some device simulations.^11,20–26 However, resistive switching models are also developed considering field driven diffusion of pre-existing defects only, without defect generation.^27–29 Experimentally, definitive evidence for one view over the other has proven extremely challenging to obtain, and the uncertainty that remains presents an obstacle to addressing pressing technological issues, such as controlling cycle-to-cycle and device-to-device variability, increasing endurance, and improving reliability.

In this article, we provide insight into this fundamental issue by investigating defect generation processes using first principles theoretical approaches. In particular, we consider TiO₂, which is a useful model system as it is both a candidate material for ReRAM and finds many other applications where non-equilibrium defect generation under electrical bias may be relevant (e.g., electrodes in rechargeable batteries, gas sensors, photocatalysts, and solar-cells^30–32). The equilibrium properties of a wide range of defects in TiO₂ have been addressed theoretically using both semi-empirical and density functional theory based approaches.^14,33–50 Here, we determine the lowest energy pathway to creation of a vacancy defect on both oxygen and titanium lattice sites, with the displaced ion occupying a neighboring interstitial site. The predicted activation barriers associated with generation of the initial vacancy-interstitial defect pair are 7.3 eV (oxygen) and 3.3 eV (titanium). Furthermore, we show that the activation barrier to formation of an oxygen vacancy defect can be reduced to 4.9 eV by trapping of holes which may be injected by the electrode. These relatively high activation energies suggest that diffusion of pre-existing oxygen vacancies (characterized by an activation energy of about 2.6 eV (Refs. 51–54)) is much more likely than defect generation in TiO₂. However, both oxygen and titanium defect creation processes are associated with generation of a net dipole moment which may serve to reduce the activation barrier under very high local electric fields (for example, as may be present during the set operation). These results provide much needed atomistic models of defect generation processes in a metal oxide material and provide key parameters needed to assess the role they play in resistive switching.

The rest of this article in organized in the following way. In Sec. II, we detail the first principles methods; we employ to predict the properties of defects in TiO₂. In Sec. III, we present our results, including the structure and formation energies of intrinsic defects in TiO₂, the structure and activation energies associated with formation of vacancy-interstitial defect pairs, and the effect of charge trapping on defect formation. Finally, in Sec. IV, we discuss the results and implications in detail before presenting our conclusions.

The properties of defects in TiO₂ are described using density functional theory (DFT) and the projector augmented wave method as implemented in the VASP code.^55,56 The Perdew, Burke, and Ernzerhof (PBE) approximation is employed for exchange and correlation,⁵⁷ and electronic states are expanded in a plane wave basis with an energy cutoff of 500 eV. To correct the self-interaction error associated with localized electrons and holes, DFT + U corrections implemented within the Dudarev scheme are applied on both O 2p and Ti 3d states.⁵⁸ We employ values of U that have been shown to reproduce spectroscopic properties of TiO₂ defects and also ensure good linearity of the total energy with electron occupation as required by a self-interaction free functional (U_Ti = 4.20 eV and U_O = 5.25 eV).⁵⁹ Using this approach, the optimized lattice parameters of rutile TiO₂ (space group P42/mnm) are obtained as a = b = 4.670 Å and c = 3.034 Å and the single electron band gap is 2.4 eV. These predicted lattice constants are within 2.5% of experiment while the band gap is underestimated by about 0.6 eV.^60,61

All defect calculations are performed using a 3 × 3 × 4 supercell containing 72 Ti atoms and 144 O atoms, and a 2 × 2 × 2 Monkhorst-Pack grid is used for k-point sampling. The stability of intrinsic defects are characterized by calculation of the formation energy

In this equation, $Edefq$ is the total energy of the supercell containing a defect in relative charge state q and E_ideal is the total energy of the ideal bulk supercell. Δn_i is the difference between the number of atoms of species i in the defective and ideal supercells, μ_i is the chemical potential of species i, and E_F is the electron Fermi energy. Total energies are corrected for potential alignment and image charge interactions as described in Ref. 62.

In order to estimate activation energies for defect formation, we first perform a linear interpolation of atom coordinates between the ideal bulk supercell (⁠$riideal$⁠) and the defective supercell (⁠$ridef$⁠),

where t is a parameter ranging from 0 (ideal crystal) to 1 (defective). We then perform a constrained minimization where the diffusing ion forming the vacancy is held fixed but all ions within a spherical shell of radius 6 Å are fully optimized. The minimized total energy of a set of configurations spanning range t = 0 − 1 represents an upper bound to the minimum potential energy surface for defect formation. This approach was adopted as attempts to employ the nudged elastic band method, which in general is a more rigorous approach,⁶³ met issues with convergence.

Some of the defects discussed in this article induce a net dipole moment relative to the bulk crystal. We estimate this dipole moment using the optimized coordinates from the constrained minimization described above $riopt(t)$ and the ionic charges obtained by Bader analysis of the corresponding charge density⁶⁴ Q_i(t). The induced dipole moment is calculated as

where the sum is restricted over ions within a spherical region of radius 6 Å centered on the defect.

Before presenting results on defect formation, we summarize the calculated structure and formation energies of the intrinsic vacancies and interstitial defects in TiO₂, which can be considered as the building blocks of the vacancy-interstitial pair defects discussed in Sec. III B. We consider both Ti and O vacancies and intersitials representing the four key intrinsic defects in the TiO₂ crystal. We perform a series of structural optimizations starting from different initial geometries and spin densities in order to determine the most stable structure for each defect type and charge state. Defect formation energies are calculated using μ_O = E(O₂)/2 (i.e., half the energy of an oxygen molecule) corresponding to oxygen rich conditions. Once μ_O is fixed, the chemical potential of Ti is also defined since $μTiO2=μTi+2μO$⁠. We note that since the DFT + U approach is parameterized, to describe defects in TiO₂ rather than the oxygen molecule formation energies defined in this way may suffer from a systematic error. In principle, this could be corrected for by comparison to experiment; however, we do not attempt such a correction here since the main focus of this study is on defects which maintain the stoichiometry of TiO₂ and so their stability does not depend on the choice of the reference chemical potential.

The most stable structure of each defect is shown in Fig. 1 with corresponding defect formation energies summarized in Table I. The dependence of the formation energy on the electron Fermi energy for all intrinsic defect types is shown in Fig. 2. In the following, we discuss the properties of each of the defects in detail with comparison to previous results wherever available.

The neutral oxygen vacancy (V_O in Kröger-Vink notation) is predicted to have a formation energy of 4.0 eV with a triplet ground state (S = 1) involving two electrons localized on neighboring Ti ions consistent with recent results obtained by electron paramagnetic resonance⁶⁵ (Fig. 1(a)). This result is also in agreement with numerous previous studies using a range of methods such as GGA + U, HSE, and sX (with formation energies ranging from 4.4–5.7 eV).^{41–43,45,46} Also consistent with previous studies, we find that only the q = 0 and q = +2 charge states are stable across the entire range of Fermi energies, characteristic of a ‘negative-U’ defect (Fig. 2). The charge transition level (i.e., the Fermi energy for which the q = 0 and q = +2 have equal formation energy) is located 0.65 eV below the conduction band minimum. This is consistent with recent results obtained using the non-local screened exchange functional.⁴⁵ 

The neutral oxygen interstitial (O_i) defect is predicted to adopt a dimer configuration by displacing one of the lattice oxygen ions. The defect can be considered as an $O22−$ species formed through the bonding of the neutral oxygen interstitial with a lattice oxygen ion which has a formal charge of −2 (i.e., $O0+O2−→O22−$⁠). This species which is not stable in the gas phase is stabilized in the TiO₂ lattice by the electrostatic environment and is characterized by an O-O bond length of 1.49 Å. This prediction is in good agreement with previous studies;⁴⁵ however, our exhaustive search for the most stable defect configuration has revealed two possible neutral oxygen interstitial dimer configurations. One in which the axis of the O-O bond is aligned with respect to the local crystal symmetry (which we refer to as planar) and one in which the dimer axis of the O-O bond is slightly rotated (which we refer to as tilted). We find that these two configurations have similar formation energies (within 70 meV) but with the tilted dimer being the ground state with a formation energy of 2.4 eV (Fig. 1(b)). We estimate that the barrier to rotate the dimer between the planar to the tilted orientation is around 100 meV, suggesting both configurations may be readily accessible under typical conditions. The oxygen interstitial is able to trap an additional one or two electrons (leading to a corresponding increase in O-O bond length to 1.90 and 2.15 Å), consistent with previous studies.^46,66 For the negatively charged oxygen interstitial defects, only the planar dimer structure is found to be stable. Similar to the case for the oxygen vacancy, only the q = 0 and q = −2 charge states are thermodynamically favored over the full range of electron Fermi energies (Fig. 2). Again, the agreement between these predictions and non-local screened exchange functional results is extremely good.⁴⁵ 

The neutral titanium vacancy (V_Ti) is predicted to have a formation energy of 5.0 eV and adopts a S = 2 spin configuration with four holes localized on neighboring O ions (Fig. 1(c)). We are not aware of any previous studies of the Ti vacancy in rutile TiO₂, but the predictions are qualitatively similar to results obtained for the same defect in anatase TiO₂.⁶⁷ As the charge of the defect is decreased from q = 0 to q = −4, each of the holes surrounding the Ti vacancy is removed one by one reducing the net spin from S = 2 to S = 0. These charge transitions involve the sequential filling of hole states predominantly localized on a single oxygen ion providing a good indication that the self-interaction error in these calculations is small. The charge transition levels are clustered around 0.5 eV above the valance band maximum (Fig. 2).

The titanium interstitial defect (Ti_i) is predicted to occupy an octahedral position (Fig. 1(d)) and can exist in either a 3+ or 4+ charge state consistent with previous studies.⁴⁵ Further electrons added to the defect localize on nearby Ti lattice sites and so are better described as weakly bound electron polarons rather than true defect charge states. The formation energies and charge transition levels predicted are again in good agreement with previous calculations⁴⁵ (Fig. 2).

The results presented in this section demonstrate that the DFT + U method we employ is able to describe the properties of all four types of intrinsic defect in rutile TiO₂ in very reasonable agreement with previous calculations employing higher level methods such as non-local hybrid functionals. Importantly, the DFT + U method is much less computationally expensive than hybrid functional approaches, allowing us to perform calculations on relatively large supercells, which is important for modeling the defect formation processes discussed in Sec. III B.

With the structure of intrinsic defects in rutile TiO₂ determined, we now turn to address the possibility of defect formation in the perfect lattice. In particular, we consider processes where an atom is displaced from its lattice site into a neighboring interstitial site (i.e., the creation of a Frenkel defect). Since vacancy and interstitial defects are stable in a number of charge states in rutile TiO₂, there are different possibilities for the Frenkel defect formation which can conveniently be expressed in Kröger-Vink notation as follows:

We note that since formation of a Frenkel defect involves no net change in the number of ions or electrons in the system, the corresponding formation energy does not depend on the electron or oxygen chemical potential. The numbers given in the square brackets above are the Frenkel defect formation energies for the reactions calculated by summing the formation energies of the individual intrinsic defects (Table I). Calculated in this way, these formation energies correspond to the formation of vacancy and interstitial pairs that are sufficiently separated that there is negligible interaction between them. For both oxygen and titanium, the most stable Frenkel defect is comprised of the intrinsic defects with the highest charge, $|q=2|$ in the case of oxygen (E_f = 6.1 eV) and $|q=4|$ in the case of titanium (E_f = 6.7 eV). These results are comparable to previous results obtained using the PBE functional.⁶⁸ Importantly, in the presence of a non-equilibrium electric field, the stability of Frenkel defects comprised of oppositely charged interstitial and vacancy defects would be enhanced and could provide a driving force for defect generation in films under electrical bias.

While the calculations above are helpful for identifying the most likely Frenkel defects to form in TiO₂, they do not allow us to say, anything about the mechanism of defect formation. To address this, we consider the stability of metastable configurations involving closely separated vacancy and interstitial defects and characterize the activation energies associated with defect generation. First, we perform a systematic search of possible vacancy-interstitial configurations in order to determine the metastable configurations with the lowest formation energy. For oxygen Frenkel defects, we consider as initial geometries the optimized structures of the q = 0, q = − 1, and q = −2 oxygen interstitial with an oxygen vacancy created in lattice sites up to the seventh nearest neighbor. For the titanium Frenkel defects, we consider the optimized structure of the q = +4 and q = +3 titanium interstitial with Ti vacancies created in lattice sites up to the second nearest neighbor. Following structural optimization, many of these initial geometries “relax” to the bulk crystal structure, indicating the lack of any nearby potential energy minimum; however, a number of metastable configurations are obtained.

Following our search, we find that the most stable oxygen Frenkel defect involves a q = −2 oxygen interstitial and a q = +2 oxygen vacancy located at the fourth nearest neighbor lattice site (E_f = 5.97 eV). The most stable titanium Frenkel defect consists of a q = +4 interstitial and a q = −4 vacancy at the neighboring lattice site (E_f = 3.15 eV). The structures of both defects are shown in Fig. 3. The fact that the most stable defects involve vacancies and interstitials with the highest charge is consistent with the result obtained for isolated defects above. However, the formation energies are in both cases lower as a result of mutual electrostatic and elastic interactions between the vacancy and interstitial. The binding of the interstitial to the vacancy is small for oxygen but high for the titanium defect due to the smaller distance between the vacancy and interstitial (1.82 Å versus 3.85 Å).

With the most stable oxygen and titanium Frenkel defect configurations identified, we calculate the activation barriers associated with the initial stage of defect formation using the constrained minimization procedure outlined in Sec. II. Fig. 4 shows the potential energy surface for both defects from which we estimate activation energies of 7.3 and 3.3 eV for oxygen and titanium, respectively. As noted above, the binding energy of the titanium interstitial to the vacancy is higher than for oxygen and further separation of the vacancy and interstitial requires overcoming an additional activation barrier of 3.4 eV. To assess the possible role of a non-equilibrium electric field in assisting the defect formation process, we calculate the corresponding induced dipole moment $|p|$ (Fig. 4). The dipole moment increases in a monotonous fashion to about 5.3 eÅ and 4.0 eÅ for the oxygen and titanium defect, respectively. In the presence of an electric field, these induced dipole moments will reduce both the defect formation energy and barrier to formation. For example, for both the O and Ti Frenkel defects shown in Fig. 4, the dipole moment close to the transition state is about 3 eÅ, and for electric fields of 1, 5, and 10 MV/cm, the expected reductions in the activation energy (E.p) are 0.03, 0.15, and 0.30 eV, respectively. We return to discuss this point in more detail in Sec. IV.

We now extend the calculations above to consider whether the trapping of electrons or holes which may be injected under a non-equilibrium bias can assist the formation of defects. This is an issue that has seen considerable speculation, but there is little quantitative known. Modeling the non-equilibrium injection and trapping of charge directly is an extremely challenging problem requiring approaches that can bridge the timescales associated with electron transfer and ionic motion.⁶⁹ However, to provide some insight, we consider a simpler model where we assess the formation energies associated with creating Frenkel defects in the presence of an additional electron or hole. In particular, in Kröger-Vink notation

On the left hand side of the reactions above, the electron initially occupies a state at the conduction band minimum and the hole occupies a state at the valence band maximum.

As in Sec. III B, for each of the reactions above, we have identified the most stable defect involving a vacancy and interstitial at close separation. For titanium, we find little reduction in defect formation energy by addition of electrons or holes. However, for oxygen defects, addition of a hole leads to a significant reduction in formation energy. In particular, we identify the following reaction as the most stable:

with a formation energy of 4.70 eV. The defect is qualitatively different to the one formed in the absence of a hole as it involves a neutral oxygen interstitial and a positively charged vacancy (Fig. 5(a)). The much lower formation energy (4.70 eV compared to 5.97 eV) can be understood in terms of the energy gained by localization of the hole.

The potential energy surface associated with defect formation is calculated in the same was as described previously (Fig. 5(b)). The estimated activation energy is 4.9 eV, considerably lower than the 7.3 eV obtained without the hole present. This is explained by the fact that in the latter case the displaced oxygen carries a −2 charge and therefore experiences stronger electrostatic repulsion with surrounding lattice oxygen ions during the transition, whereas in the former case, it is neutral. Again, we also calculate the corresponding induced dipole moment $|p|$⁠, which reaches a maximum of 2.3 eÅ.

We start with a discussion of some of the factors which may affect the accuracy of the calculations presented in this article. The approximations used in the DFT implementation, in particular, the use the DFT + U approach, may leave residual self-interaction error which will affect the formation energy of defects and the nature of charge localization. The band gap is also underestimated using this approach. However, the properties of intrinsic defects calculated in Sec. III A are in very good agreement with a range of alternative self-interaction corrected methods suggesting the approximations employed are reasonable. Another potential source of error is the use of periodic boundary conditions which leads to artificial interactions between the defect and its periodic images. To limit these effects, we have used a relatively large supercell (216 atoms) and the high dielectric constant of rutile TiO₂ means electrostatic interactions are screened effectively. Finally, another possible source of error is the constrained minimization approach employed to characterize the potential energy surface associated with defect formation. This approach provides a reliable upper estimate of the activation barrier but cannot exclude lower energy pathways involving more complex ionic displacements (although we were unable to find any after a thorough search).

We now turn to discussion of some of the potential implications of our results. Considering the possibility of electric field assisted defect generation in TiO₂, we have identified that the initial step in the formation of a charged vacancy and interstitial pair is associated with activation barriers of 7.3 eV (oxygen) and 3.3 eV (titanium). The dipole moment associated with these defects is about 4 to 5 eÅ; therefore, under an electric field, effective activation energies may be reduced. Considering the electric fields that may realistically exist in devices (1–30 MVcm⁻¹), this field effect can only offer a reduction of the order 1 eV at most. After activation, the binding between the vacancy and interstitial is relatively weak in the case of oxygen, and so the charged interstitial would be free to diffuse away driven by the electric field. For the titanium defect, the binding is stronger and a subsequent barrier of 3.4 eV must be overcome before the Ti ion is free to diffuse away. Typical activation energies associated with oxygen vacancy diffusion in TiO₂ are of the order 2.6 eV.^51–54 Even considering the effect of hole trapping which can reduce the activation energy to oxygen vacancy generation, these results suggest that diffusion of pre-existing oxygen vacancies is always much more facile than defect generation in bulk TiO₂. However, activation energies near the electrode interface^70–72 or at grain boundaries^12,73,74 may be considerably lower. Of the defect generation processes considered, Ti Frenkel formation appears as the most favorable. This is especially true in the presence of a high local electric field, suggesting this process may play a role in the set process where a high field may exist between the filament apex and the electrode.

One of the reasons the results presented in this article are useful is that there is little experimental information available on defect generation process which may take place under electrical bias in TiO₂ and oxides more generally. On the other hand, this makes direct validation of the predicted defect creation pathways and activation energies extremely difficult. One of the few examples where an activation energy for Frenkel defect generation has been extracted experimentally is in a study of forming in HfO₂ films.¹¹ In this work, the temperature dependence of the forming process was investigated and fitted to a trap-assisted-tunneling simulation including field enhanced Frenkel defect generation. An activation energy attributed to oxygen Frenkel defect formation of 4.4 eV was obtained. Although HfO₂ is a quite different material with a much wider gap, we note that this activation energy is broadly similar to the energies obtained for TiO₂ (4.9 eV and 7.3 eV).

In summary, we have performed a first principles theoretical investigation into non-equilibrium defect generation processes in TiO₂. We have identified atomistic pathways for the initial step in the generation of oxygen and titanium Frenkel defects and have characterized the corresponding activation energies. Our key conclusions are that the formation of both oxygen and titanium Frenkel defects induces a net dipole moment, and therefore, their formation can be assisted by an electric field. In the case of oxygen, we also find that trapping of holes can further reduce the barrier to defect formation. However, the activation energies remain relatively high. The lower barrier to formation of titanium defects and the larger associated dipole moment suggests such processes may be relevant in very high local electric fields (for example, as may be present during a set operation). These results provide much needed atomistic models of defect generation processes in metal oxide materials and provide key parameters needed to assess the role they play in resistive switching.²⁸ More generally, the results are relevant to a far wider range of TiO₂ applications where non-equilibrium electric field driven defect generation may contribute to material degradation, for example, in photocatalysts, dye-sensitized solar cells and electrodes in rechargeable batteries.^30–32 

K.P.M. acknowledges support from EPSRC (EP/K003151) and COST Action CM1104. This work made use of the facilities of Archer, the UK's national high-performance computing service, via our membership in the UK HPC Materials Chemistry Consortium, which is funded by EPSRC (EP/L000202). All data created during this research are available by request from the University of York Research database http://dx.doi.org/10.15124/0bede454-b78e-474c-8d93-38b9911700f5.