1. Introduction

Recently the conduction mechanism of resistive random access memory (RRAM) has been drawing extensive research. Many researchers have used the conductive filament model to explain the resistive switch behavior of the RRAM. For most conduction filament-type RRAM devices, rupture and the formation of localized conduction filaments (CFs) is the main physical mechanism of resistive switching behaviors. The CFs, which may consist of either an intrinsic oxygen vacancy (V$_{\rm O})$ from crystal or metal ions resolving from an electrode^[1-3], plays a crucial role in the resistive switching behavior of the RRAM. However, the RRAM devices often show a large variation in switching voltage due to the random growth/rupture of the CFs^[4]. This random characteristic often makes it difficult for the CFs to form or break along the same path in each switching cycle. Therefore, in order to improve the uniformity of operation voltage, a controllable mechanism of CF growth should be established. Several works have been done to address this problem. Some research has improved the controllability of CFs by a thin GST interface layer^[5], metallic nanocrystal (NC)^[6], lighting rod effect^[7] and doping effects^{[8, 9]}. From previous studies, V$_{\rm O}$ properties in the functional layer material of RRAM devices such as HfO$_{2}$^[10] or Al doped HfO$_{2}$^{[11, 18, 20]} have been addressed. The formation energy of V$_{\rm O}$ ($E_{\rm V_O}^{\rm f})$ is significantly influenced by doping trivalent ions^{[8, 9]}. Up to now, however, some researchers have mainly focused on the relationship between $E_{\rm V_O}^{\rm f}$ and dopants characters, such as chemical valence and dopant ionic radius^{[12, 13]} in doped materials. They neglected the relationship between $E_{\rm V_O}^{\rm f}$ and the distance between V$_{\rm O}$ and dopants. Moreover, after V$_{\rm O}$ being formed near the dopant, there will be an interaction between the dopant and charge-compensating vacancy to form a distinct defect cluster^{[12, 14]}. Fewer researchers have found a relationship between the interaction of defects and electrical properties of RRAM devices. Interactions between oxygen vacancies also exist, as found in NiO^[15]. In a Ni/NiO RRAM device, the formation and rupture of an oxygen vacancy chain is due to the presence of V$_{\rm O}$ interactions which affect the RS characters^[16]. In this paper, we adopt the first principle method to confirm doping localized effects and discuss the relationship between defect interactions and electrical properties of an Al-doped HfO$_{2}$ RRAM device. From another aspect we could understand the relationship of material properties and device electrical characteristics and it provides guidance for the optimization design of RRAM device material. We focus on the structural and energetic aspects in this paper. A detailed discussion on the electronic structures will be given in a separate paper.

2. Material and methods

In this study, the first-principle calculations are based on the density-functional theory (DFT) within the Perdew Burke Ernzerhof (GGA-PBE) for the exchange-correlation potential, and Vanderbilt type ultrasoft pseudopotentials for core-valance interaction, as implemented in the CASTEP code^[17]. The valence configurations of the pseudopotentials are 5p$^6$5d$^2$6s$^2$ for Hf, 2s$^2$2p$^4$ for O. The plane wave set basis cutoff energy is 340 eV, $k$-points sampling is performed at 2 $\times$ 2 $\times$ 2 special points in a Monkhorst-Pack grid. Geometries of supercells are fully relaxed until atomic forces are smaller than 0.03 eV/Å. A jellium background is used to neutralize the lattice in the calculations for charged defects. To calculate the relative parameters, a supercell (2 $\times$ 2 $\times$ 2) of monoclinic HfO$_{2}$ containing 96 atoms is constructed. The Coulomb interaction between charged defects in different periodic cells is calculated using the Makov-Payne scheme, and due to the large size of the supercell separating the periodic defect images by over 10 Å, the corresponding image charge correction is below 0.1 eV. Oxygen atoms are removed from the supercell in order to model the vacancy defects, one doping Al atom replaces the Hf site in this supercell to build a doping system, which is energetically favorable under oxygen-rich conditions^[18]. Corresponding to the above, it is equivalent to implanting Al atoms forming about 1% concentration into the HfO$_{2}$ layer.

3. Results and discussions

The formation energy of charged V$_{\rm O}$ is defined as^[19]

$E_{\rm f} (X^{Q})=E_{\rm tot} (X^{Q})-E_{\rm tot} ({\rm bulk})+Q(\varepsilon_{\rm f} +\varepsilon_{\rm v} +\Delta V)+\mu _{\rm O},$

(1)

where $E_{\rm tot}(X^{Q})$ and $E_{\rm tot}$(bulk) are the total energy of the supercell containing $X^{Q}$ with charge $Q$ and the supercell without the $X^{Q}$, respectively. $\varepsilon_{\rm f}$ is the Fermi level, referenced to the edge of valence band (VB) $\varepsilon_{\rm v}$, $\Delta V$ is the shift of $\varepsilon_{\rm v}$ due to the introduced charge of $X^{Q}$ defects in the supercell. $\mu_{\rm O}$ is the chemical potential of oxygen, and it is half of the oxygen molecule energy in this calculation. Due to the low concentration of Al, V$_{\rm O}^{+1}$ is chosen to be calculated for study doping effects^[20]. The $\varepsilon_{\rm f}$ is set as 0 eV when the formation energy of V$_{\rm O}^{+1}$ is calculated.

As we know, there are two types of oxygen vacancies in monoclinic hafnia: threefold and fourfold-coordinated (V$_{\rm O3}$ and V$_{\rm O4}$, respectively)^[10]. As shown in Fig. 1, due to the asymmetry of the monoclinic HfO$_2$, there may be different distances between one dopant and two types of oxygen vacancies in the same position of NN (nearest neighbor), NNN (next-nearest neighbor) or even as far as NNNN (next-next-nearest neighbor) positions.

As shown by the computed results in Fig. 2, trivalent Al has significantly reduced both $E_{\rm V_O}^{\rm f}$ of V$_{\rm O3}^{+1}$ and V$_{\rm O4}^{+1}$, compared with the un-doped HfO$_2$. This is due to the formation of Hf-O-Al^[20]. It is noticed in Fig. 2(a) that the formation energy of V$_{\rm O3}^{+1}$ and that of V$_{\rm O4}^{+1}$ at the NN position to the Al are both smaller than that at other positions. With the distance increasing, the doping effect on the $E_{\rm V_O}^{\rm f}$ will be weakened, which indicates that the dopant will enhance the formation of V$_{\rm O}$ which at nearly positions. Compared with the doping effect, the formation energy of V$_{\rm O3}^{+1}$ and V$_{\rm O4}^{+1}$ which are at different distances from the other neutral V$_{\rm O3}$ in pure HfO$_2$ are also calculated, as shown in Fig. 2(b). It is found that as the distance increases, the deviation of $E_{\rm V_O}^{\rm f}$ with respect to distance does not exceed 1%. This indicates one V$_{\rm O}$ has no obvious influence on the formation of the other V$_{\rm O}$.

In this paper, the achieved stability of CFs is due to the localization effect of dopant. The stability of V$_{\rm O}$ not only depends on its formation energy, but relates to the interaction between the dopant and V$_{\rm O}$. The interaction with V$_{\rm O}$ should also be taken into account. These interactions should not be neglected when we investigate the RS behaviors of RRAM devices. The interaction energy ($E_{\rm int})$ is calculated when two defects are in different distances away, the general relationship is the following:

$E_{\rm int} =E_{\rm cluster} -\sum\limits_{\rm component} {E_{\rm isolate-defect}}.$

(2)

The computation results are shown in Fig. 3. It is known that two defects with opposite charge: Al$_{\rm Hf}^{-1}$ and V$_{\rm O}^{+1}$/V$_{\rm O}^{+2}$, form the (Al$_{\rm Hf}$-V$_{\rm O}$)$^0$ and (Al$_{\rm Hf}$-V$_{\rm O}$)$^{+1}$, respectively^[18]. In this paper, we choose the neutral complex (Al$_{\rm Hf}$ -V$_{\rm O})^0$ to study the interaction energy between the dopant and V$_{\rm O}$. Here we show $E_{\rm int}$ between Al$_{\rm Hf}^{-1}$ and V$_{\rm O}^{+1}$, the vertical axis denotes $E_{\rm int}$. The lower $E_{\rm int}$ is, the more the configuration of Al$_{\rm Hf}^{-1}$ and V$_{\rm O}^{+1}$ is stable. The horizontal axis represents the relative position of Al$_{\rm Hf}^{-1}$ and V$_{\rm O}^{+1}$. It is shown in Fig. 3(a) that V$_{\rm O}$ is stable at the first nearest O sites of Al$_{\rm Hf}^{-1}$. With increasing distance, the dopant's localized effect will be weakened, as the case of $E_{\rm v_o}^{\rm f}$. It may be concluded that V$_{\rm O}$ is effectively trapped at these sites due to an attractive coulombic interaction between Al$_{\rm Hf}^{-1}$ and V$_{\rm O}^{+1}$, or local structural modifications^[21] induced by dopants of different sizes compared to the host Hf$^{+4}$ ions would be another possible reason to account for the defect interaction energy. Recently it has been shown that V$_{\rm O}^{+1}$ and V$_{\rm O}^0$ will stabilize a cohesive filament in TiO$_{2}$-based RRAM devices^[22]. Then the $E_{\rm int}$ between two oxygen vacancies with different charge states in HfO$_2$ was calculated. In Figs. 3(b) and 3(c), some interaction energies between oxygen vacancies are negative although repulsive interactions are expected between defects with the same charge states. With increasing distance, the $E_{\rm int}$ of the V$_{\rm O3}$-V$_{\rm O3}$ pair has obviously changed while $E_{\rm int}$ of V$_{\rm O3}$-V$_{\rm O4}$ has a small deviation. The obvious differences of the two types of V$_{\rm O}$ pairs may be due to different influences from structural modifications or electron interactions. The maximum interaction energy is achieved when two V$_{\rm O}$'s are singly ionized. These may be due to atomic rearrangements and hybridization between vacancy orbitals^[23]. A similar effect has been observed in NiO^[15-16] as well.

To further investigate the influence of defect clusters on the characteristics of doped HfO$_2$, the PDOS of each different configurations has been calculated, as shown in Fig. 4. From the PDOS analysis, the gap between defect level and conduction-band minimum (CBM) in Fig. 4(a) is smaller than that in Fig. 4(b), respectively, indicating that when V$_{\rm O}$ is trapped by the dopant at the adjacent position, the defect level will move upward. Based on the physical model of bipolar resistive switching behavior established by Gao^[24], it will cause the reduction of the critical electrical field to reset the devices and result in faster reset speed of RRAM devices^[9]. Since Figure 5 has shown the PDOS of V$_{\rm O}$-V$_{\rm O}$ pairs in pure HfO$_2$, the gap between the defect level and the conduction band minimum has no obvious change, however, when V$_{\rm O}$ at the NN site of the other V$_{\rm O}$, the peak position of the O(p) orbital has a deviation of 0.092 eV from the peak position of Hf(d) orbital due to the electronic interaction between these orbitals. When the oxygen vacancies form a cluster, the defect states overlap and energy levels are split due to bonding or antibonding combinations of wave functions^[15]. From band structure analysis which is not shown in this paper, two oxygen vacancies create two defect energy levels within the band gap region, as the same results that oxygen vacancy clusters may introduce a subband in the gap^[25].

Figure 6 shows the different location of the two oxygen vacancies with respect to the Al cation. It is clear that the configuration corresponding to Fig. 6(a) has the lowest energy (-39936.97 eV) due to defect interactions, indicating that the oxygen vacancy clusters will be stable near the dopants. Comparing Fig. 6(b) (-39937.28 eV) and Fig. 6(c) (-39937.16 eV), it also confirms that the oxygen vacancies tend to form clusters in doped HfO$_2$ which is used as the function layer of RRAM devices.

As discussed above, it can be concluded that there is a relationship between the defect interaction and the process of "forming" and "set/reset" in the Al-doped RRAM device, which has been illustrated in Fig. 7. This interaction has two effects on the device resistive switch behaviors. First, during the working state, voltage is applied to the whole dielectrics and it enhances the generation of oxygen vacancies. From analysis above, oxygen vacancies will spontaneously generate and form clusters close to the dopants due to the interaction of V$_{\rm O}$-V$_{\rm O}$ and Al-V$_{\rm O}$ pairs, as shown in Fig. 7(a). With the help of clusters, consequently, the CFs will be steadily formed around the Al-doping sites by linking these clusters, as shown in Fig. 7(b). The distribution of the forming voltage ($V_{\rm forming})$ and set voltage ($V_{\rm set})$ of the RRAM device will be improved due to this stable process, which is confirmed by experimental results as shown in Figs. 8(b) and 8(c). Then a better uniformity of $V_{\rm forming}$ and $V_{\rm set}$ has been achieved^[8] Second, during "forming" and each"set/reset"cycle, fewer V$_{\rm O}$ are needed to link the dissolved Vo clusters, as shown in Fig. 7(c). This will mean that the strength of the electric field that can generate V$_{\rm O}$ becomes lower. This inference is also confirmed by experimental results as shown in Figs. 8(a) and 8(c) in which the $V_{\rm forming}$ and $V_{\rm set}$ are both reduced^[8].

4. Conclusion

Based on the first principle methods, it is proved that the doping effects can improve the uniformity of oxide-based RRAM device from the perspective of defect interactions. The dopants have localization effects on not only formation energy but interactions of oxygen vacancies in the neighbourhood. These intrinsic defects tend to form clusters due to interactions between V$_{\rm O}$-V$_{\rm O}$ and dopant-V$_{\rm O}$ pairs. For localized CFs, these clusters will be helpful for improving the stability of the growth/rupture process. Therefore, they will optimize the resistive switching process of RRAM. This paper provides another perspective to understand the physical mechanism of doping effects on oxide-based RRAM devices.