The exact composition and structure of conductive filaments in hafnia-based memristors are still not fully understood, but recent theoretical investigations reveal that hexagonal HfOx phases close to the h.c.p. Hf structure are probable filament candidates. In this work we list h.c.p. Hf, Hf6O, Hf3O and Hf2O as possible phases for the filament in hafnia memristors. Their differences in lattice parameters, electronic structures and O charge states are studied in details. Migration of O ions for both in-plane and out-of-plane directions in these phases is investigated using first-principles calculations. Both single-phase supercells and filament-in-dielectric models are used for migration barrier calculations, while the latter is proven to be more accurate for the c-direction. The migration of O ions is fastest in metal Hf, while slowest in Hf2O. The existence of O interstitials in Hf tends to hinder the transport of O.

Binary oxide-based memristors1,2 that demonstrate stable resistive switching phenomenon3,4 are considered as promising candidates for next-generation non-volatile memory in microelectronics. Among them, more attention has been paid to HfOx, TaOx, TiOx and AlOx,5–15 due to their excellent technical compatibility to CMOS flow-line. When active electrodes such as Ag and Cu are not involved, their resistive switching phenomena belong to the valence change mechanism.3 In this case the movement of O anions and the formation of oxygen-deficient conductive filaments (CFs) are the common features that account for their resistive switching, but the exact composition of the CF is specific to each memristor material. For example, the Magnéli phase crystal TinO2n-1 has been identified as the CF structure in TiOx-based memristors,16 while for TaOx memristors the CF composition was found to be TaO1-x,17 and probably amorphous.18 When it comes to HfOx, which has been intensively studied as a very promising and reliable memristor material, the exact CF composition is unfortunately less known. Recent experimental investigations agree on the point that the CF in HfOx consists of O vacancies,19–24 but the composition (or composition range) of the CF was not given explicitly. Hence, it is still a challenging task to directly identify the CF composition and structure of HfOx-memristors from transmission electron microscopy and scanning tunneling microscopy.25 

On the other hand, theoretical investigation on the reduction process of HfO2 may provide useful information about the CF composition in HfOx memristors. Although the thermodynamically stable HfOx (0 ≤ x ≤ 2) phases at room temperature only involve monoclinic HfO2 (m-HfO2), hexagonal Hf6O and hexagonal Hf,26 several intermediate phases have been predicted through first-principles calculations.27–30 Upon reduction, the structure of HfOx basically experiences a transition from tetragonal to hexagonal symmetry as shown in Figure 1. While the initial composition HfO2 points to a monoclinic P21/c phase, it can become tetragonal (a distorted fluorite structure with space group P42/nmc) when heated up to ∼2000 K.31,32 Except for HfO2, the other HfOx compositions follow the tetragonal-to-hexagonal trend. For instance, the ground state of Hf2O3 has been predicted to be tetragonal P4¯m2,27 and the ground state of HfO was predicted by Zhu et al. as tetragonal I41/amd.30 Nevertheless, another hexagonal phase of HfO (h-HfO), following P6¯2m ZrO,28 is also close in energy.29 Subsequently, for HfO1-x several trigonal/rhombohedral structures have been known, including Hf2O, Hf3O and Hf6O. These phases can simply be regarded as hexagonal metal Hf with certain amount of O interstitials, thus in this work we categorize them into a global hexagonal HfOx class. All these sub-oxide phases with HfO1.5-x stoichiometry are metallic or with very tiny band gap (as in the case of h-HfO29). Yet, according to the thermodynamic calculation of McKenna,33 as well as the O vacancy chain cohesion study from Xue et al.,34 it is reasonable to attribute the CF in HfOx memristors to those hexagonal phases close to metal Hf, such as Hf2O, Hf3O, Hf6O and Hf.

FIG. 1.

Structure and symmetry of various HfOx phases during the reduction process of HfO2. The stoichiometry for the tetragonal-to-hexagonal turning point is HfO.

FIG. 1.

Structure and symmetry of various HfOx phases during the reduction process of HfO2. The stoichiometry for the tetragonal-to-hexagonal turning point is HfO.

Close modal

Nevertheless, the difference among various hexagonal HfOx phases as the CF is still not known. In particular, during the so-called RESET operation, which corresponds to the rupture of the filaments by surrounding O anions as illustrated in Figure 2 (SET is the opposite process to turn the device into low-resistance state), O movement is at the heart of the switching mechanism.35 On the one hand, the pre-existence of O interstitials in the CF may hinder the further transport of O anions inside the CF. On the other hand, these O interstitials may also expand the volume of the CF to facilitate O anion transport therein. Understanding the special characteristics of various HfOx CF candidates is beneficial for experimental identification of the true CF composition and structure. In this work, we focus on the migration of O anion inside or across these CF candidates, and will calculate the various diffusion barriers in these processes, which can afford valuable information on the CF formation and rupture kinetics in HfOx memristors.

FIG. 2.

Movement of O anions during the RESET process of a typical TiN/HfO2/Pt memristor cell.

FIG. 2.

Movement of O anions during the RESET process of a typical TiN/HfO2/Pt memristor cell.

Close modal

We carried out density functional theory calculations using the plane-wave basis set and the projector augmented-wave method, as implemented in the Vienna Ab initio Simulation Package (VASP).36,37 A constant 500 eV plane-wave kinetic energy cutoff was used throughout the calculations. For exchange-correlation energy, the generalized gradient approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE) was utilized.38 The valence electron configurations are 5d and 6s for Hf, as well as 2s and 2p for O. The Brillouin zones were sampled using Γ-centered k-point mesh. For HfxO supercells, a 5×5×3 mesh was used, while only the Γ point was used for filament-in-dielectric models. Diffusion barriers were calculated using a climbing image nudged elastic band (CI-NEB) method, developed by Henkelman and H. Jónsson.39 

In Table I we list the optimized lattice constants of h.c.p. Hf, h-Hf6O, h-Hf3O, h-Hf2O and h-HfO. As their unit cells contain different numbers of Hf atoms, we here define a set of effective lattice parameters a’ and c’ for better comparison. These parameters are based on the unit cell of h-Hf6O, which has 18 Hf atoms in total. Hence, in other materials a’ and c’ are the supercell lattice parameters, such that the supercell also contains 18 Hf atoms. As illustrated in Figure 3, both a’ and c’ increase monotonously from Hf to Hf3O. After adding more O interstitials to reach the Hf2O stoichiometry, however, only c’ increases, rather than a’, but the variations are so tiny that Hf2O may still be categorized into the same class as Hf6O and Hf3O, as they are all based on h.c.p. Hf. The P6¯2m phase of h-HfO is, however, topologically quite different from the other phases as shown in Figure 1. Consequently, it demonstrates a great shrink in the a’ value, but in the mean time a very large expansion along the c-direction.

TABLE I.

Lattice parameters of various HfOx phases in the hexagonal class, obtained with GGA-PBE calculations. The effective parameters a’ and c’ correspond to those of the supercells containing 18 Hf atoms.

Optimized lattice parameters
Composition a (Å) a’ (Å) c (Å) c’ (Å) 
Hf 3.191 5.527 5.069 15.207 
Hf65.545 5.545 15.336 15.336 
Hf35.571 5.571 15.355 15.355 
Hf25.562 5.562 5.148 15.445 
HfO 5.226 5.226 3.172 19.035 
Optimized lattice parameters
Composition a (Å) a’ (Å) c (Å) c’ (Å) 
Hf 3.191 5.527 5.069 15.207 
Hf65.545 5.545 15.336 15.336 
Hf35.571 5.571 15.355 15.355 
Hf25.562 5.562 5.148 15.445 
HfO 5.226 5.226 3.172 19.035 
FIG. 3.

Variation of the effective lattice parameters upon inserting O interstitials into metal Hf: (a) a-axis; (b) c-axis. The stoichiometries here cover Hf, Hf6O, Hf3O, Hf2O and HfO.

FIG. 3.

Variation of the effective lattice parameters upon inserting O interstitials into metal Hf: (a) a-axis; (b) c-axis. The stoichiometries here cover Hf, Hf6O, Hf3O, Hf2O and HfO.

Close modal

An investigation into the electronic structures shows that Hf6O, Hf3O and Hf2O possess similar density of states (DOS) around the Fermi level as metal Hf, and all phases are good conductors (see Figure 4). However, the bandwidth of the O sub-band strongly depends on the stoichiometry. A very narrow O sub-band emerges slightly below -7 eV (with respect to the Fermi level, sic passim) in Hf6O, which is broadened substantially in Hf3O. Moreover, in Hf3O the upper O sub-band edge moves to around -6.5 eV. For Hf2O, the O sub-band becomes even broader, whose upper edge further increases a bit. The trend is reasonable since ultimately in HfO2, the valence band maximum will be dominated by states from O, rather than Hf.40 

FIG. 4.

(a) Electronic density of states (DOS) of h.c.p. metal Hf; (b) DOS and partial DOS of Hf6O; (c) DOS and partial DOS of Hf3O; (d) DOS and partial DOS of Hf2O.

FIG. 4.

(a) Electronic density of states (DOS) of h.c.p. metal Hf; (b) DOS and partial DOS of Hf6O; (c) DOS and partial DOS of Hf3O; (d) DOS and partial DOS of Hf2O.

Close modal

Subsequently, we focus on the charge state of the O atoms in these HfOx phases. To this end we calculated their Bader charges in the primitive cells. As summarized in Table II, in all phases O atoms are in perfect -2 valency, i.e., O2- anions. Hence, it is in general not possible to achieve neutral O atoms inside the CF of hafnia-based memristors. The charge status of O has been fixed by the Hf environment, which is very similar to a recent finding that the Ag charge status is determined by some solid-state electrolyte environments.41 To further strengthen this point, we have set up HfOx supercells (with at least 72 atoms), and introduce an extra O interstitial, either in the neutral O atom form, or in the O2- anion form. The following Bader charge calculation confirms that there is almost no difference on the charge status of this extra O interstitial (Table II). Note that there is only one kind of O site in Hf6O and Hf3O, but two in the case of Hf2O. The site named O-I is in the sparse interval layer that contains less O atoms, while the site named O-II is in the dense interval layer of Hf2O.

TABLE II.

List of calculated O Bader charges inside various HfOx phases. In the primitive cell calculation, the values refer to the O atoms in the perfect stoichiometry. For supercell calculations, the values refers to that of an extra O interstitial introduced, either neutral or with two extra electrons (the “2e” case).

Number of charges on O
according to the Bader gauge
Interstitial in
Primitive cell supercell
CompositionO-IO-IINeutral2e
Hf -2.10 -2.10 
Hf6-2.09 -2.06 -2.06 
Hf3-2.09 -2.03 -1.99 
Hf2-2.04 -2.06 -1.96 -1.96 
Number of charges on O
according to the Bader gauge
Interstitial in
Primitive cell supercell
CompositionO-IO-IINeutral2e
Hf -2.10 -2.10 
Hf6-2.09 -2.06 -2.06 
Hf3-2.09 -2.03 -1.99 
Hf2-2.04 -2.06 -1.96 -1.96 

After identifying the delicate differences among the fundamental properties of these CF candidates, we then calculate the O anion migration barriers within these phases. Since the fundamental properties of h-HfO deviate from the Hf—Hf6O—Hf3O—Hf2O series, such as in the trend of lattice constant, we here focus only on Hf, Hf6O, Hf3O and Hf2O. For each phase, a supercell of around 1 nm × 1 nm × 1.5 nm was set up to carry out NEB calculations. The migration paths are illustrated in Figure 5, and the migration barriers are listed in Table III as well as illustrated in Figure 6. Two trends are clearly observed. First, the in-plane migration barriers (a-b or b direction) are enlarged when more O are intercalated into the Hf lattice. Take the b-direction as an example. The migration barrier follows the sequence of Hf (1.97 eV) < Hf6O (2.26 eV) < Hf3O (2.36 eV) < Hf2O (2.51 eV). This implies that the pre-existing O atoms inside Hf tend to hinder the in-plane O migration, whose effect overwhelms the lattice constant expansion. Secondly, the vertical migration barriers along c-axis are generally much larger than the corresponding in-plane migration barriers. We also find that in most cases, as an extra O interstitial, the migration barriers get lowered compared with moving an existing O atom in these CF candidates.

FIG. 5.

Migration paths under investigation for (a) O interstitial inside Hf; (b) O inside Hf6O; (c) O interstitial inside Hf6O; (d) O inside Hf3O; (e) O interstitial inside Hf3O; (f) O inside Hf2O. The starting points of the paths are marked by red arrows.

FIG. 5.

Migration paths under investigation for (a) O interstitial inside Hf; (b) O inside Hf6O; (c) O interstitial inside Hf6O; (d) O inside Hf3O; (e) O interstitial inside Hf3O; (f) O inside Hf2O. The starting points of the paths are marked by red arrows.

Close modal
TABLE III.

List of migration energy barriers of O ions inside Hf, Hf6O, Hf3O and Hf2O, calculated using supercells with fixed dimension.

Migration barriers
Along a-bAlong bVertical (along c)
 Hf 
O (Interstitial) 1.99 eV 1.97 eV 3.12 eV 
 Hf6
O (Original) 2.27 eV 2.26 eV 3.47 eV 
O (Interstitial) 2.06 eV 2.05 eV 3.05 eV 
 Hf3
O (Original) 2.37 eV 2.36 eV 3.33 eV 
O (Interstitial) 2.50 eV 1.96 eV 2.85 eV 
 Hf2
O (Original) 2.50 eV 2.51 eV 3.64 eV 
Migration barriers
Along a-bAlong bVertical (along c)
 Hf 
O (Interstitial) 1.99 eV 1.97 eV 3.12 eV 
 Hf6
O (Original) 2.27 eV 2.26 eV 3.47 eV 
O (Interstitial) 2.06 eV 2.05 eV 3.05 eV 
 Hf3
O (Original) 2.37 eV 2.36 eV 3.33 eV 
O (Interstitial) 2.50 eV 1.96 eV 2.85 eV 
 Hf2
O (Original) 2.50 eV 2.51 eV 3.64 eV 
FIG. 6.

Energy profiles during the CI-NEB calculations for the transport of (a) O interstitial inside Hf; (b) O inside Hf6O; (c) O interstitial inside Hf6O; (d) O inside Hf3O; (e) O interstitial inside Hf3O; (f) O inside Hf2O.

FIG. 6.

Energy profiles during the CI-NEB calculations for the transport of (a) O interstitial inside Hf; (b) O inside Hf6O; (c) O interstitial inside Hf6O; (d) O inside Hf3O; (e) O interstitial inside Hf3O; (f) O inside Hf2O.

Close modal

One should, however, be cautious about the migration barrier results calculated using fixed supercells, since the imposed geometry constraints can be too strong to imitate the experimental situation. To further estimate the accuracy of our migration barriers, we studied the transport of O ions in “filament-in-dielectric” models that seem more practical from an experimental point of view. For Hf-in-HfO2 model (417 atoms), we established a partial CF structure such that the migration barriers both in the lateral direction and in the vertical direction can be calculated (Figure 7 ab). A more complicated Hf6O-in-HfO2 model with 526 atoms was also set up as in Figure 7 cd, where the CF is continuous due to the large lateral area and size limitation. To cover more possibilities of migration, for some barriers we calculated two distinct paths. The lateral migration barrier of an O ion from m-HfO2 to metal Hf was estimated to be 1.73 eV and 1.86 eV along two different paths (one of them illustrated in Figure 7 a), both very close to the lateral migration barrier inside pure Hf (1.97 eV – 1.99 eV). On the other hand, the vertical entrance into metal Hf suffers from a barrier of 1.79 eV to 1.82 eV (see Figure 7 b), which is substantially lower than that inside metal Hf calculated previously along c-axis (3.12 eV, obtained using the supercell approach). To exclude the influence of interface penetration, we further calculated the migration barrier of this intercalated O ion inside Hf along its c-axis, and found only a 1.50 eV migration barrier (less than a half of the value calculated using the supercell approach). In the case of Hf6O-in-HfO2, the lateral migration barrier was estimated to be 2.04 eV and 2.44 eV following two paths, respectively (one of them illustrated in Figure 7 c). The vertical barrier is 1.82 eV following the path in Figure 7 d, which is also much smaller than that inside pure Hf6O (3.05 eV).

FIG. 7.

Migration paths for O from the dielectric region to the CF region. (a) Lateral migration from m-HfO2 to Hf; (b) vertical migration from m-HfO2 to Hf; (c) lateral migration from m-HfO2 to Hf6O; (d) vertical migration from m-HfO2 to Hf6O.

FIG. 7.

Migration paths for O from the dielectric region to the CF region. (a) Lateral migration from m-HfO2 to Hf; (b) vertical migration from m-HfO2 to Hf; (c) lateral migration from m-HfO2 to Hf6O; (d) vertical migration from m-HfO2 to Hf6O.

Close modal

A paradox then arises as the migration barriers calculated using supercell approach and filament-in-dielectric approach do not fully agree with each other. While for in-plane barriers they support each other, for vertical migration barriers the supercell approach predicts much higher energy barriers than the values obtained using ultra-large filament-in-dielectric models. It needs to be pointed out here that the latter approach is more accurate, while the supercell approach over-estimated the barriers along c-axis, as explained below. The a-b plane is a close-pack plane in metal Hf. When O ions migrate along c-axis, they must penetrate these dense atomic planes made of Hf. As the supercell dimensions were fixed during NEB calculations, this is quite difficult since the Hf close-pack plane can hardly show large in-plane deformation. However, in the latter approach, there are Hf/HfO2 interfaces involved, which can be subject to compression more easily. In experiments, a CF is limited in lateral size and its interfaces with the surrounding dielectric can indeed be compressed to allow for vertical O penetration. Therefore, the lower vertical migration barriers, such as the 1.82 eV barrier in Hf6O, make more sense than the supercell-obtained values (3.47 eV, or 3.05 eV if in the foreign interstitial form). On the other hand, the accurate in-plane migration barriers calculated using the supercell approach can also be understood from the special h.c.p. structure of metal Hf. As the c-direction is perpendicular to the close-pack plane, Hf atomic planes could demonstrate deformation in this direction easily. When O migrates in the gap region between two Hf atomic planes, vertical displacement of adjacent Hf atoms naturally occurs to ensure the transport even in fixed supercell format. After considering the limitation of the fixed supercell approach, we conclude that in these CF candidates the migration of O ions along c-axis is at least as easy as the in-plane directions. In any direction, the typical migration barrier is less than 3 eV.

The mechanism discussed above may also be used to explain some over-estimated theoretical migration barriers in the literature. For instance, the diffusion barrier of O in h.c.p. Ti was calculated to be 3.5 eV by Traoré et al. using the supercell approach, which was regarded as too high.42 It can be more realistic to use a multi-grain model so as to allow for the necessary interfacial deformation.

Our calculation results imply that O migration is possible for both in-plane and out-of-plane directions for h.c.p. Hf-like CFs. Experimental evidences are still lacking regarding whether it is the c-axis or the a/b axis of the metal that is perpendicular to the capacitor area. However, in any case our results show that O migration can occur both vertically and horizontally during RESET and SET operations. The vertical movement directly leads to the rupture or re-formation of the CF at its weak location, while the horizontal movement is relevant to the shrink/growth of the CF lateral dimensions.

The reduction process of HfO2 generally involves a lattice symmetry transformation from tetragonal to hexagonal, where the transition point lies at HfO. Hence, h.c.p. Hf, Hf6O, Hf3O and Hf2O are potential candidate phases for the conductive filament in HfO2-based memristors. Bader charge calculation confirms that the O interstitials in these phases are in perfect O2- anion form, which precludes the possible existence of neutral O atoms inside metal Hf. The difficulty of O ion migration in these phases depends on the concentration of exiting O interstitials. The in-plane migration barrier along b-axis increases from 1.97 eV in the case of Hf, to 2.51 eV in the case of Hf2O. Hence, the existing O ions tend to hinder in-plane migration of O, such that pure metal Hf is the fastest media for O transport in the series of filament candidates considered in this work. The vertical migration barriers are over-estimated by the fixed supercell approach, but using larger filament-in-dielectric models we obtained typical migration barriers along c-axis as 1.50 eV and 1.82 eV in Hf and Hf6O, respectively. Our calculation supports fast migration of O anions around the Hf/Hf6O filaments in HfO2-based memristors, regardless of the direction of transport, but adding more O interstitials in the filament tends to hinder the O transport.

This work was financially supported by the National Key Research and Development Plan of the MOST of China under Grant No. 2016YFA0203800, the National Natural Science Foundation of China under Grant No. 11704134 and No. 51732003, the Fundamental Research Funds of Wuhan City under Grant No. 2017010201010106, and the Hubei “Chu-Tian Young Scholar” program.

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