Solid-electrolyte-based resistive random switching memory (ReRAM) based on transition metal oxides (TMOs), also known as electrochemical metallization (ECM) cells, is a very attractive candidate to replace flash memory due to its simple metal– insulator– metal (MIM) sandwich structure, ^[1] high density, long retention time, excellent scalability, low power consumption, ^{[2– 5]} and compatibility with CMOS technology. Among a wide variety of TMOs^{[6– 19]} (HfO₂, ZrO₂, SiO₂, TiO₂, Ta₂O₅, etc.) that have been proposed thus far, HfO₂ is particularly attractive due to its possible role in replacing silicon dioxide as the gate dielectric in a CMOS logic transistor and also due to its good resistive switching (RS) capability.^{[7– 9]} Generally, the ECM device consists of a thin film of solid electrolyte (‘ I’ ), an active electrode(such as Ag, Cu, or Ni), and an inert electrode(such as Pt, W, or Au). When a positive voltage is applied to the active electrode (e.g., Ag), Ag⁺ ions are generated and will migrate onto the inert electrode, and then they will be reduced into Ag atoms on the surface of the inert electrode. CFs will grow from the cathode toward the anode. Vice versa, the CF will be dissolved due to oxidation when a voltage of opposite polarity is applied.^[10] It is widely accepted that electrochemical metallization is the main mechanism that is responsible for the formation and rupture of conducting filaments (CFs).^{[11– 13]} However, the ReRAM devices often show a large fluctuation in switching voltages due to the random growth or rupture of the CFs.^[14] More recently, the study of defect migration path has aroused the considerable interest of researchers. Knowledge of the migration orientation in advance is an effective way to improve characteristics of ReRAM. For example, the operation voltage can be lowered along the optimized orientation and reduce power consumption. Wang et al.^[15] have investigated the optimal migration pathway of oxygen vacancies in TiO₂ by using the nudged elastic band method. Zhu et al.^[16] have studied the diffusion path of oxygen vacancies in TiO₂ by using the nudged elastic band method. As described by Tingkun Gu’ s model, the conductive path in Cu-based Ta₂O₅ along [001] orientation has been observed by studying the isosurface plot of the partial charge density.^[17] Lu et al.^[18] discussed the optimal migration path of Cu in HfO₂ along the [001] direction, which is helpful for reducing power consumption in HfO₂ based ReRAM. Liu et al.^[19] have observed a conductive filament in Ag-based SiO₂ within the (111) plane by using transmission electron microscopy (TEM). At the same time, most experiments have confirmed that metal conductive filaments form by the Ag atom.^{[20, 21]} However, there is no thorough understanding of the optimal filament direction of Ag-related metal dopants in the resistive switching material.

In reality, considering the experimental limitations such as producing cost and manufacturing time, software simulation and different methods for theoretical analysis are used to explore the performance parameters such as oscillating orientation and migrating direction of Ag atoms in HfO₂-based ReRAM on a nanoscale. In this paper, we theoretically investigate the migration path of Ag in HfO₂-based ReRAM device by employing first-principles theory. Firstly, with three analytical tools (molecular dynamic simulations, migration barriers and population analysis), the dominant motion of the Ag atom (at sites 1 and 3) in HfO₂ supercells is studied. Subsequently, the optimal long-arrange migration path of Ag in HfO₂ supercells is also investigated.

The CASTEP code^[22] with Vanderbilt ultrasoft pseudopotentials is used to perform all the calculations apart from those determining migration barriers. The Perdew Burke Ernzerhof generalized gradient approximation (GGA-PBE)^[23] is used to describe the electronic exchange-correlation potential. After the energy convergence test, the plane-wave cutoff energy is set to be 400  eV, and a grid of 2× 2× 2 Monkhorst– Pack k-piont is used for Brillouin zone integrations. The valence electrons are assumed to have 2s²2p⁴configuration (O), 5d²6s² configuration (Hf), and 4d¹⁰5s¹ configuration (Ag). Monoclinic HfO₂^[24] is used with the lattice parameters of a = 5.117  Å , b = 5.1754  Å , and c = 5.2915  Å . The supercell geometries are fully relaxed until atomic forces are smaller than 0.03  eV/Å , while the lattice vectors of supercells are fixed during geometry optimization. The molecular dynamics calculations^{[25, 26]} are performed with a simulation at room temperature of 300  K. A Nosé thermostat and no pressure are adopted. The total simulation time is 5  ps in time steps of 1  fs, leading to a total number of steps of 5000. The first 2.5-ps time is used in order to reach system balance, and the following 2.5-ps time is used to obtain statistical position coordinates of the Ag atom. The model in this work utilizes periodic boundary condition (PBC) in order to ensure that the system does not have an abrupt border with a vacuum. In order to consider the influences of temperature and pressure on ion diffusion, the NPT ensemble is used, and the periodic Ag images are over 5  Å away from each other. This reduces the Coulomb interaction between images of the Ag in different periodic cells.

The DMol³ code^{[27, 28]} with a complete-LST/QST search protocol is used in order to calculate migration barrier of the Ag atom in supercell. The position corresponding to the highest point of energy (i.e., the transition site) can be obtained by calculating the migration energy, which can directly reflect the degree of difficulty of Ag migration. This is an approach used successfully in previous studies on cation migration in complex oxides.^{[15, 16, 29, 30]}

In order to qualitatively measure the strength of covalent bonding between atoms, the overlap population between atoms is calculated by Mulliken’ s population analysis method.^[31] In Mulliken analysis, the charge Q(A) associated with a given atom A, and bond overlap population n(AB) between A and B atoms are defined respectively as

where P_{μ ν} (k) and S_{μ ν} (k) are the density matrix and the overlap matrix, respectively, and W_k is the weight associated with the calculated k-point in the Brillouin zone. The n(AB) can be used to assess the covalent strength of a bond. A value of zero indicates a perfectly ionic bond, while values greater than zero indicate increasing the levels of covalency.^[32] Positive and negative values indicate bonding and antibonding states, respectively.

There are three possible interstitial sites (1, 2, and 3) and one possible substitutional site for Ag atom in the unit cell of HfO₂ as shown in Fig.  1. It was reported by Zhou et al.^[33] that the favorable site for Ag atom is the site 3 (the formation energy is 5.9  eV) and the formation energy of Ag at sites 1 and 2 are 6.53  eV and 6.6  eV, respectively. However, the calculations in this work show that the formation energies of Ag at the sites 1 and 2 are both 6.54  eV, which means the two positions are equivalent. Thus, in the following work, the properties of Ag atoms at sites 1 and 3 in HfO₂ are discussed.

Fig.  1.

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	Fig.  1. Unit cell of HfO2. The numbers 1, 2, and 3 represent the possible interstitial sites, and the number 4 refers to the possible substitutional site. The smaller balls represent O atoms, and the bigger balls denote the Hf atoms.

Figure  2(a) shows the supercell (2× 2× 2) model with four Ag atoms located at site 3 within four unit cells. Site 3 is set as an initial site within the simulation. The thermal motion of Ag atoms is simulated at room temperature (300  K) by using the molecular dynamics method, and the position coordinates of Ag atoms are collected for every other 50 steps, after sufficient relaxation of the structure. The results show that there is a dominant vibrating trend of Ag atoms along the direction as displayed in Fig.  2(b), and the effective distances of the circled Ag along this orientation is approximately 1.9416  Å .

Fig.  2.

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Fig.  2. (a) Smaller balls represent O atoms, middling balls denote the Ag atoms and bigger balls denote the Hf atoms. A supercell (2× 2× 2) containing 100 atoms with four Ag in each of four unit cells at position 3. (b) Thermal motion position coordinates of Ag atoms at the position 3 after sufficient relaxation of the structure. A cluster refers to the corresponding thermal motion position coordinates of one Ag atom.

Examination of Ag atom motion in HfO₂ is of vital interest when using it as the anode material in RRAM. Simulation can greatly enhance our understanding of the defect process or migration path by evaluating the activation energies. Since long-range migration of Ag is very difficult, we only discuss hoping between the nearest unit cells. Five main migration possibilities of Ag atoms at site 3 are considered within m-HfO₂ (illustrated in Fig.  3(a)). Furthermore, the specific migration barriers of Ag atom along the five diffusion paths are calculated. The migration barrier profiles each as a function of path coordinate are shown in Fig.  3(b), and the transition states (saddle point) of Ag atom along these paths are also marked out (balls marked with number 2) in Fig.  4. For paths A, B, and C, the Ag atom, which is located on site 3 in the unit cell, migrates to the equivalent site 3 of another neighboring unit cell along the [100], [010], and [001] directions, respectively. For pathD and path E, the Ag atom, which is located at site 3 in the unit cell, migrates to site 1 of another neighboring unit cell along the and [110] directions, respectively. By employing the DMol³ method as shown in Fig.  3(b), for diffusion path B, the energy is found to display a single maximum, corresponding to a saddle point at the path coordinate 0.48 between the initial site and the final site with a migration energy of 6.62  eV. It is obvious that path B consumes more energies than any other orientation for the migration of the Ag atom. However, the migration barrier of path D (3.02  eV, the path coordinate 0.48) is lower than those of path A (6.21  eV, the path coordinate 0.54), path C (3.29  eV, the path coordinate 0.67), and path E (6.95  eV, the path coordinate 0.53), which is possibly due to diverse electron distributions and interactions between Ag atoms and other atoms along these paths. The above descriptions combined with thermodynamic analysis further indicate that Ag atoms tend to migrate along the orientation.

Fig.  3.

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	Fig.  3. (a) Five different diffusion paths of Ag atoms (at site 3) in HfO2. Paths A, B, C, D, and E are along the [100], [010], [001], , and [110] directions, respectively. (b) Energy profiles of Ag atom along the five paths with respect to its energy in the initial structure. The highest points of curves refer to the highest values of energy along the five diffusion paths.

Fig.  4.

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Fig.  4. Barrier sites of Ag atoms along the five paths. Smaller balls represent O atoms, bigger balls denote the Hf atoms, and middling balls refer to the Ag atoms. For each of paths A, B, and C, site 3 on the left, site 2, and site 3 on the right are the initial site, transition site and the final site of Ag atom, respectively. For each of paths D and E, site 3 on the left, site 2, and site 1 on the right are the initial site, transition site and final site of Ag atom, respectively.

The thermodynamic analysis and migration barrier of Ag atom show that the path from site 3 to site 1 of the adjacent cell is the optimal migration path for Ag atom in HfO₂. Thus an interesting question arises: which direction is best for the migration of Ag atoms at site 1. Next, the thermodynamic motion and migration barrier of Ag atom at site 1 will be discussed. A supercell (2× 2× 2) model with four Ag atoms located at site 1 in each of four unit cells is built as displayed in Fig.  5(a). The simulation method of the thermal motion is the same as that of Ag atoms at site 3. The result shows that there is a vibrating trend for Ag along the [001] direction as shown in Fig.  5(b), and the effective distance between the circled Ag atoms along this orientation is about 1.1269  Å .

Fig.  5.

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Fig.  5. (a) Smaller balls represent O atoms, middling balls denote the Ag atoms, and bigger balls refer to Hf atoms. A supercell (2× 2× 2) containing 100 atoms with four Ag atoms in each of four unit cell at position 1. (b) The thermal motion position coordinates of Ag at the position 1 after sufficient relaxation of the structure. A cluster refers to the corresponding thermal motion position coordinates of one Ag.

In order to further confirm the migration direction of Ag atoms at site 1, the migration barrier of Ag atom along the possible diffusion paths are calculated. As illustrated in Fig.  6(a), we also choose five main migration path candidates characterized by their corresponding migration rates of paths A, B, C, D, and E. For paths A, B, and C, the Ag atom, which is located at site 1 of the unit cell, migrates to the equivalent site 1 of another neighboring unit cell along the [100], [010], and [001] directions, respectively. For pathD, the Ag atom, which is located at site 1 in the unit cell, migrates to the symmetrical site 1 of the neighboring unit cell along the direction. For path E, the Ag atom, is located at site 1 of the unit cell, migrates to site 3 of the neighboring unit cell along the [101] direction. The most favorable migration path among the five migration paths is determined by comparing their migration barriers. By employing the DMol³ method, the migration barrier profiles versus path coordinates are obtained as shown in Fig.  6(b). The barrier sites of Ag atoms along these paths are also marked out (balls marked with number “ 2” ) in Fig.  7. For diffusion path C, it shows a point of maximum energy, corresponding to a saddle point almost located in a midway between the initial site and the final site with a migration barrier of 7.62  eV (path coordinate of 0.46). Obviously, path D shows the low migration barrier (4.59  eV, path coordinate of 0.50) compared with those along path A (5.58  eV, the path coordinate of 0.53), path B (4.86  eV, the path coordinate of 0.48), and path E (6.11  eV, the path coordinate of 0.58). It signifies that the direction energetically gives the optimal route of Ag among the five paths.

Fig.  6.

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	Fig.  6. (a) Five different diffusion paths of Ag in HfO2. Paths A, B, C, D, and E are along the [100], [010], [001], , and [101] directions, respectively. (b) Energy profiles of Ag atom at site 1 along the five paths with respect to the energy of the initial structure. The highest points of curves refer to the highest values of energy along the five diffusion paths.

Fig.  7.

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Fig.  7. The barrier sites of Ag atom along the five paths. Smaller balls represent O atoms, bigger balls denote the Hf atoms and middling balls refer to the Ag atoms. For each of paths A, B, C, and D, site 1 on the left, site 2, and site 1 on the right are the initial site, transition site and final site of Ag atom, respectively. For path E, site 1 on the left, site 2, and site 3 on the right are the initial site, transition site and final site of Ag atom, respectively.

However, the thermodynamic analysis shows that Ag atoms tend to move along the [001] path, which is in disagreement with the migration barrier calculations. The final motion direction is perhaps further affected by the diverse electron distributions and interactions between the Ag atom and other atoms when the Ag atom can go as far as possible. As presented in the following section, the population analysis indicates that it will be more favorable for Ag atoms to migrate along the direction.

Fig.  8.

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Fig.  8. Smaller balls represent the O atoms, middling balls denote the Ag atoms, and bigger balls refer to the Hf atoms. (a) A supercell (2× 2× 2) containing 100 atoms with four Ag atoms in each of four unit cell at the symmetrical site 1. (b) The thermal motion position coordinates of Ag atom at the position 1 after sufficient relaxation of the structure. A cluster refers to the corresponding thermal motion position coordinates of one Ag.

The further movement direction of Ag at the symmetric site 1 is also investigated. Figure  8(a) shows the built supercell (2× 2× 2) model with four Ag atoms located at the symmetric site 1 in each of four unit cells. The simulation method is the same as that for Ag atoms at site 3. The result shows that there is a vibrating trend for Ag atom, which is along the [001] direction as displayed in Fig.  8(b), and the effective distance between the circled Ag along this orientation is about 0.8529  Å .

The migration barriers of Ag atom along possible diffusion paths are calculated. As illustrated in Fig.  9(a), we also choose five main migration path candidates characterized by their corresponding migration rates for paths A, B, C, D and E. For paths A, B, and C, the Ag atom, which is located at site 1 in the unit cell, migrates to the equivalent site 1 of another neighboring unit cell along the [100], [010], and [001] directions, respectively. For pathD, the Ag atom, which is located at site 1 in the unit cell, migrates to site 3 of the neighboring unit cell along the direction. For path E, the Ag atom, which is located at site 1 in the unit cell, migrates to the symmetrical site 1 of another neighboring unit cell along the [110] direction. Employing the DMol³ method, the migration barrier profiles versus path coordinates are obtained as shown in Fig.  9(b). The barrier sites of Ag atom along these paths are also marked (balls marked with number 2) in Fig.  10. The result shows that migration barrier of path D (3.21  eV, the path coordinate 0.44) is lower than path A (4.87  eV, the path coordinate 0.53), path B (5.04  eV, the path coordinate 0.48), path C (7.58  eV, the path coordinate 0.45), and path E (6.12  eV, the path coordinate 0.51). However, the optimal migration path that is calculated by migration barrier has a  slight disagreement with the molecular dynamics calculation, which may be similar to the case of Ag at site 1. Population analysis of the Ag– O bond at the transition state is performed in the following section.

Fig.  9.

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	Fig.  9. (a) Five different diffusion paths of Ag in HfO2. Paths A, B, C, D, and E are along the [100], [010], [001], , and [110] directions, respectively. (b) Energy profiles of Ag atom at position 1along the five paths with respect to the energy of the initial structure. The highest points of curves refer to the highest values of energy along the five diffusion paths.

Fig.  10.

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Fig.  10. Barrier sites of Ag atom along the five paths. Smaller balls represent the O atoms, bigger balls denote the Hf atoms, and middling balls refer to the Ag atoms. For each of paths A, B, and C, the site 1 on the left, site 2, and site 1 on the right are the initial site, transition site, and final site of Ag atom, respectively. For each of paths D and E, site 1 on the left, site 2, and site 3 on the right are the initial site, transition site, and final site of Ag atom, respectively.

Furthermore, we calculate the overlap population values between Ag atom and its surrounding atoms of the above three migration steps in transition states in five diffusion paths. The results are shown in Table  1. The S₃ (site 3) → S_fin represents that the Ag atom migrate from initial site 3 to final site, and a similar meaning can be applied to S_S1 (symmetrical site 1) → S_fin and S₁ (site 1) → S_fin. The calculated average overlap population values of Ag– Hf bonds of the transition states are negative, revealing that an anti-bonding characteristic is induced between Ag and Hf shells. Thus, a great deal of attention has been paid to the characteristic of bonding in this work, and the overlap population between Ag– O bonds is only discussed. It can be seen that the average overlap population values of Ag– O show relatively low values in three different cases: S₃ → S_fin with an overlap population value of 0.02 (along the direction), S_S1 → S_fin with an overlap population value of 0.11 (along direction), and S₃ → S_fin with an overlap population value of 0.03 (along direction). A value for the overlap population close to zero indicates that there is no significant interaction between the electronic populations of the two atoms. This indicates that the bonding interaction between Ag and O at the above three migration steps in the direction is weaker than those in the other four directions, which further indicates that it is more favorable in energy for Ag atoms to migrate at the transition state in the direction.

Table 1.

Table 1.

Table 1. Population analyses of Ag– O bonds of the transition states along [100], [010], [001], , [110], and [101] directions, respectively. Orientation Overlap population S3 → Sfin SS1 → Sfin S1 → Sfin [100] 0.23 0.13 0.21 [010] 0.21 0.18 0.19 [001] 0.05 0.22 0.20 0.02 0.11 0.03 [110] 0.12 – 0.22 [101] – 0.21 –

Table 1. Population analyses of Ag– O bonds of the transition states along [100], [010], [001], , [110], and [101] directions, respectively.

The lowest-energy pathway for long-range diffusion is obtained through analyses of thermodynamic calculations, the migration barriers, and the bond overlap populations. The diffusion loop path Γ ₁ → Γ ₂ → Γ ₃ → Γ ₁ for Ag atoms are marked as shown in Fig.  11. Migration paths from site 3 to site 1, from site 1 to the symmetrical site 1 and from the symmetrical site 1 to site 3 are marked as step Γ ₁ (along ), step Γ ₂ (along ), and step Γ ₃ (along ), respectively.

Fig.  11.

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	Fig.  11. Schematic diagram of the transmission path of Ag and sandwich structure in a HfO2-based ReRAM. For HfO2 slab, smaller balls represent the O atoms, bigger balls denote the Hf atoms.

Ultimately, the lowest migration barrier path of Ag in HfO₂ supercells is built by considering the results of the molecular dynamic simulations, migration barriers, and population analysis. As viewed from different directions, we can see that the Ag migrate along [111] or directions, as shown in Figs.  12(a)– 12(d).

Fig.  12.

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	Fig.  12. Optimal migration paths of Ag in HfO2 viewed from (a) [010], (b) , (c) [110], and (d) directions.

In this study, we model Ag motion from two sites, i.e., site 1 and site 3, in bulk HfO₂ by using first-principles calculations. Thermodynamic analysis shows that the motion of Ag atoms in the HfO₂ supercell appears to be anisotropic, however, the migration barrier and bond overlap population calculations indicate that for Ag in HfO₂ the migration path in the direction is dominant over the other four diffusion paths. When a positive voltage is applied along this orientation, the conduction filament may form more easily, which will contribute to a faster response speed and lower power consumption in ReRAM devices. We hope that this theoretical finding is valuable for designing and fabricating the oxide-based ReRAM with a faster response speed and lower power consumption.