Memristive materials are materials that exhibit a pinched-hysteresis loop in their current–voltage characteristics.^[Reference Strukov, Snider, Stewart and Williams^1, Reference Chua and Kang^2] These materials have potential applications as non-volatile memories,^[Reference Akinaga and Shima^3] non-linear analog circuit elements,^[Reference Muthuswamy and Kokate^4] and neuromorphic circuits.^[Reference Jo, Chang, Ebong, Bhadviya, Mazumder and Lu^5] Although there are many forms of memristive materials including spin memristive systems,^[Reference Pershin and Di Ventra^6] molecular memristive system,^[Reference Agapito, Alkis, Krause and Cheng^7] ferroelectric memristive system,^[Reference Chanthbouala, Garcia, Cherifi, Bouzehouane, Fusil, Moya, Xavier, Yamada, Deranlot, Mathur, Bibes, Barthélémy and Grollier^8] and chalcogenide memristors,^[Reference Oblea, Timilsina, Moore and Campbell^9] the most actively studied form of memristive material is a two-terminal metal/oxides/metal sandwich structure. The oxide is either perovskite^[Reference Asamitsu, Tomioka, Kuwahara and Tokura^10] or binary oxide,^[Reference Strukov, Snider, Stewart and Williams^1, Reference Lee, Chen, Wu, Chen, Chen, Wang, Tzeng, Lin, Tsai and Lien^11–Reference Lee, Kim, Yun, Rhee and Yong^15] and contains a high concentration of defects such as oxygen vacancies or metal ions. This architecture has generated great interest due to its simplicity, but there are many unknowns about the fundamental mechanism of memristive behavior. It is generally agreed that the memristive behavior originates from the motion of defects. However, the details of the defect dynamics remain elusive and the hypotheses are often cited to interpret experiments without sufficient validation—especially at the atomic-scale. One widely used hypothesis is that the defects are driven by the electrostatic field and thus move in the direction of the field.^[Reference Strukov, Snider, Stewart and Williams^1]

Polycrystalline forms of binary oxides with high oxygen deficiencies, such as HfO₂,^[Reference Lee, Chen, Wu, Chen, Chen, Wang, Tzeng, Lin, Tsai and Lien^11] NiO,^[Reference Baek, Lee, Seo, Lee, Seo, Suh, Park, Park, Kim, Yoo, Chungand and Moon^12] and ZnO,^[Reference Chang, Lai, Wu, Wang, Chen and Tsai^13–Reference Lee, Kim, Yun, Rhee and Yong^15] are especially attractive memristive materials, as their fabrication is compatible with the well-established complementary metal–oxide semiconductor (CMOS) technology. It is acknowledged that the grain boundaries in polycrystalline materials play a key role during the switching process, as they are the likely places of defect aggregation and conductive path formation.^[Reference McKenna and Shluger^16] One hypothesis is that the defect motion is enhanced along the grain boundary (GB), which facilitates memristive switching.^[Reference McKenna and Shluger^16–Reference Park, Jeon, Chae, Han, Park, Seo and Kim^18] Polycrystalline ZnO thin films are widely studied for their memristive behavior. These films are grown parallel to the c-axis of wurtzite ZnO,^[Reference Chang, Lai, Wu, Wang, Chen and Tsai^13–Reference Lee, Kim, Yun, Rhee and Yong^15] and feature grain boundaries parallel to the growth axis, which is also the direction of the applied external electric field. In such configurations, the two hypotheses mentioned above seem quite credible and the pictorial scheme of defects moving along the external field (thus parallel to the GBs) is widely used to describe the experiments.^[Reference Chang, Lai, Wu, Wang, Chen and Tsai^13, Reference Yu and Wong^19]

In this paper, we propose a new mechanism of memristive switching that operates in polycrystalline ZnO and possibly other polycrystalline materials. Using results from the density functional theory calculations, we show that the motion of oxygen vacancies—the most likely defect—during switching to the conductive state is not driven by the electrostatic effect of the external field, but instead is driven by the recombination of carriers at these vacancies, which dumps energy at the defects and therefore greatly enhances their diffusion. Meanwhile, we show that the memristive behavior cannot be caused by the motion along the GB, i.e., parallel to the direction of the applied electric field. Instead, the oxygen vacancies move perpendicular to the GB, attaching and detaching from it during the switching process. The directional preference is the result of the intrinsic anisotropy of the oxide film.

The device of interest in this study is a two-terminal metal/oxide/metal sandwich structure with ZnO as the switching layer. In this layout, only devices with polycrystalline ZnO film show the switching behavior, whereby the grain boundaries play a key role. In these structures, the ZnO films are grown parallel to the c-axis of the wurtzite lattice, and consist of columnar grains.^[Reference Chang, Lai, Wu, Wang, Chen and Tsai^13] Therefore, grain boundaries parallel to the c-axis, for example, the Σ1, Σ7, Σ13, and Σ19 grains boundaries,^[Reference Korner, Bristowe and Elsasser^20] should be abundant. In this study, we pick the Σ1 GB, which has the lowest energy,^[Reference Korner, Bristowe and Elsasser^20] for the investigation. We also make the usual assumption that oxygen vacancy V _O is the defect that causes memristive switching.

Our calculations employ the Perdew–Burke–Ernzerhof^[Reference Perdew, Burke and Ernzerhof^21] version of the generalized-gradient approximation exchange-correlation functional. We use projector augmented wave potentials^[Reference Kresse and Joubert^22] and plane-wave basis as implemented in the Vienna Ab-initio Simulation Package (VASP) code.^[Reference Kresse and Furthmuller^23] The kinetic energy cutoff of the plane wave basis is set to be 400 eV. The structural relaxations are converged to 10⁻³ eV for the total energy difference between two ionic steps. The Σ1 GB is modeled by a 4 × 10 × 2 supercell which contains a pair of Σ1 grain boundaries and a total of 160 Zn and 160 O atoms. Brillouin zone sampling is performed by using the Gamma (0, 0, 0) point only. The calculations on an isolated V _O are conducted in a 2 × 2 × 2 64 atom supercell, with Brillouin zone sampling performed using a shifted 2 × 2 × 2 K-point mesh. The diffusion barriers are calculated using the nudged elastic band method.^[Reference Henkelman and Jonsson^24]

We start by examining the interaction between the V _O and the GB. It is known that the V _O in ZnO is a negative-U center that is stable in neutral and 2+ charge states (V _O⁰ and V _O²⁺).^[Reference Janotti and Van de Walle^25] We computed the binding energy of the V _O at the Σ1 GB and obtained the values of 0.1 eV and 0.2 eV for V _O⁰ and V _O²⁺ respectively. Therefore, as expected, the GB is the preferred aggregation site for oxygen vacancies.

We then examine the preferential migration direction for V _O. We first investigate the diffusion of a V _O in bulk ZnO. We found that the diffusion is very anisotropic: for the migration parallel to the c-axis, the energy barriers are 1.7 and 1.6 eV for V _O⁰ and V _O²⁺ respectively; perpendicular to the c-axis, the corresponding barriers are 1.3 and 1.2 eV. Therefore, the migration of V _O in bulk ZnO has higher barriers along the c-axis, which is parallel to the GB and the external electric field.

The GB breaks the translational symmetry in bulk ZnO and can alter the potential energy landscape of defect migration. For a defect at the Σ1 GB, there are three possible directions as shown in Fig. 1: (1) parallel to the c-axis; (2) perpendicular to the c-axis and parallel to the GB; (3) perpendicular to both the c-axis and the GB. We computed the diffusion barriers for V _O along these three directions and found that the activation energy along path 1 is 1.6 eV for both V _O⁰ and V _O²⁺; the activation energies along path 2 are 1.4 and 1.3 eV for V _O⁰ and V _O²⁺, respectively; and the activation energies along path 3 are 1.2 and 1.0 eV. Overall, although the GB modifies the diffusion of the defects, the changes are small in this case and the strong anisotropy of the bulk materials is preserved at the GB. Path 3 is the most favorable while path 1 is the least favorable. These results, combined with the binding energy of a V _O at the GB and the diffusion barriers in bulk ZnO, suggest that the attaching and detaching of V _O to the GB is more likely than the diffusion of V _O parallel to the c-axis at the GB.

Figure 1.

The possible diffusion directions for a defect at the Σ1 GB of ZnO. The large grey atom and small red atom represent Zn and O, respectively. The orange dotted line highlights the location of the GB. The purple dashed, yellow dotted, and blue filled arrows mark the three possible diffusion directions.

The next question is whether the existence of an external electric field would reduce the diffusion barrier at the GB along the c-axis and thus trigger the diffusion of V _O. We estimated the magnitude of barrier lowering using the switching electric field reported in Ref. Reference Chang, Lai, Wu, Wang, Chen and Tsai13 and found that it is smaller than 6 meV. This amount of barrier lowering can neither trigger the diffusion of V _O nor make the migration parallel to the c-axis more favorable.

The above calculations and analysis point to the conclusion that the diffusion perpendicular to the GB is favored. This conclusion holds for both the SET process—switching from high resistive state (HRS) to low resistive state (LRS) and the opposite RESET process—switching from LHS to HRS. For the RESET process, the perpendicular diffusion corresponds to the migration of oxygen vacancies away from the GB and thus the dissolution of the conductive path/filament. As the RESET process starts with the LRS, there exists a large current and significant Joule heating in the conductive filament, which could drive the diffusion of oxygen vacancies. However, for the SET process, as the current is relatively low, the effect of Joule heating at the beginning of the switching process is much smaller than the RESET process. Thus, some other mechanism must be in play.

When an external voltage is applied to the ZnO film at the HRS, it generates a small current through the Poole–Frenkel (PF) emission,^[Reference Chang, Lai, Wu, Wang, Chen and Tsai^13] which consists of two processes. The first process is the trapping of a conduction band electron by a defect, in this case, the oxygen vacancy. The second process is the emission of the trapped electron to the conduction band under the external electric field. The first process is also called “recombination” as it can be viewed as an electron recombines with a (supposedly electron-deficient) defect. In this process, the vacancy gains significant amount of energy, which is stored as the vibration of its neighbor atoms and enhances its diffusion. This is “recombination enhanced diffusion”, or more specifically, the “phonon-kick” mechanism of recombination enhanced diffusion that has been known for defects in semiconductors.^[Reference Lang^26, Reference Itoh and Stoneham^27] However, to the best of our knowledge, this is the first time that this mechanism is evoked to explain memristive switching.

In the phonon-kick mechanism, the normal diffusion barrier E is reduced by the capture energy E _R, which is defined as the total amount of energy that is released in the capture process.^[Reference Lang^26] Using results reported by Janotti and Van de Walle, (Figs 1(a) and 3(a) in Ref. Reference Janotti and Van de Walle25), we obtain an E _R = 0.5 and 1.5 eV for V _O²⁺ and V _O⁺ to capture an electron. However, these E _R values are only nominal, as (a) the direction of lattice vibration after the “phonon-kick” is not the same as the V _O diffusion, therefore, only a fraction of the nominal E _R is effective in enhancing the diffusion, (b) the defect can experience more recombination processes (kicks) before the vibrational energy from previous kicks is fully dissipated into the lattice, thus, the effective E _R may be greater than what would be expected from only one kick. Nevertheless, it is reasonable to conclude that the phonon-kick recombination process can provide enough energy to drive the diffusion of V _O at the HRS. It is interesting to note that the oxygen vacancies in oxides often exhibit strong lattice relaxations upon charge changes,^[Reference Karazhanov, Zhang, Wang, Mascarenhas and Deb^28, Reference Gavartin, Muñoz Ramo, Shluger, Bersuker and Lee^29] which makes them susceptible to phonon-kick recombination enhanced diffusion. This is probably why memristive behavior is often observed in oxides.

As more oxygen vacancies move to the GB, they inevitably interact with each other and form clusters that are essentially fragments of the conductive filament. Within these filament fragments, electronic transport is dominated by metallic conduction instead of PF emission. Meanwhile, if a filament fragment behaves as a large defect and participates in the PF emission, the wavefunction of the captured electron is delocalized over multiple vacancies. As a result, the capture energy is dissipated over more atoms, minimizing the effective E _R. Therefore, the recombination-enhanced diffusion mechanism is quenched at the GB, making these newly formed filament fragments stable during and after the SET process.

The overall schematic of the SET and RESET processes at the Σ1 GB of ZnO is shown in Fig. 2. It is as if the GB is breathing the oxygen vacancies, therefore, we term it “breathing mechanism”. There are two requirements for this mechanism. First, the responsible defect should show significant recombination-enhanced diffusion. Second, diffusion perpendicular to the GB should be favored. In the case of ZnO films, this requirement is satisfied by the intrinsic anisotropy of the bulk material. For materials other than ZnO, different factors that are intrinsic to the oxide film, such as strain or the existence of the GB itself, may also satisfy this condition. As both the SET and RESET processes are independent of the direction of the applied electric field, it is clear that this model corresponds to the unipolar switching that is often observed in polycrystalline ZnO films.^[Reference Chang, Lai, Wu, Wang, Chen and Tsai^13, Reference Lee, Kim, Yun, Rhee and Yong^15] The mechanism of bipolar switching, which is also observed in polycrystalline ZnO film,^[Reference Xu, Liu, Sun, Liu, Han, Wang, Han, Kang and Yu^14, Reference Lee, Kim, Yun, Rhee and Yong^15] is not in the scope of this study.

Figure 2.

The schematic of the “breathing mechanism” of switching. The purple circles represent oxygen vacancies. The orange dotted line represents the GB.

The identification of the switching mechanism allows us to perform realistic simulations of the defect dynamics in the memristive devices. To demonstrate this, we employed a kinetic Monte Carlo simulation of memristive switching in a ZnO film with the Σ1 GB. The simulation cell contains ~10⁴ Zn and O atoms and the oxygen vacancy concentration is set to 2%. All of the kinetic parameters such as the diffusion barriers and binding energies are from our first-principles calculations, except that an artificial value of the capture energy E _R = 0.9 eV is subtracted from the DFT diffusion barriers in the SET process in order to represent the effect of recombination. In Fig. 3, we show the SET and RESET process in a part of the simulation cell. The top panel shows the initial HRS, where the oxygen vacancies are distributed randomly in the polycrystalline ZnO film. The simulation for the SET process is run at room temperature. After 70 ns, an aggregation V _O at the GB is observed, which corresponds to the formation of the conductive filament. The simulation for the RESET process is run at a high temperature of 1150 K, emulating the condition of the filament under significant local Joule heating. After 60 µs, the filament completely dissolved and the distribution of V _O returned to an HRS. The switching behavior and time scale are consistent with the experimental data.^[Reference Chang, Lai, Wu, Wang, Chen and Tsai^13, Reference Lee, Kim, Yun, Rhee and Yong^15, Reference Kim, Moon, Gupta, Yoo and Choi^30]

Figure 3.

The kinetic Monte Carlo simulation of switching at the Σ1 GB. Only a fraction of the total simulation cell is shown. The blue dots represent the oxygen vacancies. The grey and red color represent the positions of the O and Zn atoms in the simulation cell. (RT and HT stand for “room temperature” and “high temperature”, respectively.)

In summary, using results from the first-principles calculations and combining them with the available experimental and theoretical data, we show the feasibility of a new model for memristive switching in polycrystalline ZnO. The two key ingredients in this model are (1) the SET process is triggered by recombination-enhanced diffusion; (2) diffusion perpendicular to a GB is favored due to the intrinsic properties of the oxide film. As the two conditions are quite general, we propose that this model may work in other polycrystalline oxides as well.