# 110° 110 edge dislocations in strontium titanate: Charged vs neutral, glide vs climb

Pierre Hirel, Patrick Cordier, Philippe Carrez

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Pierre Hirel, Patrick Cordier, Philippe Carrez. 110° 110 edge dislocations in strontium titanate: Charged vs neutral, glide vs climb. Acta Materialia, 2025, Acta Materialia, 285, pp.120636. 10.1016/j.actamat.2024.120636. hal-04891775

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Submitted on 23 Jan 2025

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Full length article

⟨110⟩{⟨110⟩ edge dislocations in strontium titanate: Charged vs neutral, glide vs climb

Pierre Hirel<sup>a,∗</sup>, Patrick Cordier<sup>a,b,†</sup>, Philippe Carrez<sup>a</sup>

ARTICLE INFO

Keywords:

Numerical simulation

SrTiO₃

Dislocation

ABSTRACT

The ductile deformation of strontium titanate SrTiO₃ at low temperature (T ≲ 1000 K) is commonly associated with the activity of dislocations gliding in {110} slip planes. While dislocations with pure screw character keep essentially the same dissociated core structure in the whole temperature range, dislocations with edge character may adopt different atomic configurations, associated with different charge states and different mobilities. In this work we use atomic-scale simulations to investigate the core structure of charged and neutral edge dislocations. We report a new possible dislocation core that is charge-neutral, dissociated, and with the lowest Peierls stress reported so far, thus making it an efficient component of plastic deformation. In comparison, dislocations carrying a positive charge are slightly less mobile, while those with negative charge have a very low mobility. Our results indicate that glide of charge-neutral dislocations is favoured, and that they may locally acquire a charge by interacting with vacancies all while remaining glissile. Finally, we investigate edge dislocations that are dissociated in their climb plane, and confirm that they are energetically more favourable than their glissile counterparts. Computing the activation energy for the core transformation provides insight into the ductile–brittle transition.

0. Introduction

In the early 2000s, demonstration of the plastic deformability of strontium titanate SrTiO₃ or STO at room temperature [1,2] has renewed interest in the role of dislocations in this class of materials, and paved the way for dislocation-enhanced properties of ceramics [3–5]. Since then, the mechanical behaviour of STO has been investigated both experimentally and numerically. STO exhibits a double ductile–brittle–ductile transition that is quite uncommon in ceramics. In the temperature range 0–1000 K (so-called Regime A in [2,6]), plastic deformation involves glide in {110} planes, as evidenced very early by optical micrographs [1]. At the lowest temperatures (below approx. 115 K), long straight dislocation lines are observed, mainly of pure screw character [6]. This indicates that edge segments are very mobile at these temperatures, so that dislocation loops unfold and relax into straight screw segments. In this regime, plastic deformation is therefore essentially limited by the mobility of screw segments which are dissociated [7,8]. Their motion is associated with the formation of kink pairs on partial dislocations. In a previous study we investigated the competition between simultaneous nucleation on both partials, and alternating nucleation on the leading and trailing partials, and shown

that a switch from one mechanism to the other is responsible for the
sharp increase in flow stress below 210 K [9].

As temperature increases, in particular at room temperature and above, curved dislocation lines are observed during compression of single-crystals [6], and also during nano-indentation [10,11]. These give a hint that edge segments are becoming less mobile. Compared to screw dislocations, modelling dislocations of pure edge character is more challenging due to the ionic nature of the material, which raises questions about the charge state and possible artefacts due to the choice of plane of cut. In a first study we showed that ⟨110⟩ edge dislocations are dissociated, and can glide easily through the lattice [8]. However the charge of these dislocations was not evaluated, and in retrospect it turns out they carried a positive electric charge. Marrocchelli et al. proposed a method for obtaining charge-neutral dislocations, by constructing a dipole of edge dislocations and displacing oxygen ions [12]. We implemented this solution in our study of edge dislocations in MgSiO₃ perovskite [13].

Despite this expansive body of studies, the details of dislocation core structures and migration mechanisms at the atomic scale are still disputed today. Recently Klomp et al. used atomic-scale simulations to investigate ⟨110⟩ edge dislocations in SrTiO₃ [14], and reported

<footnote><sup>a</sup> Univ. Lille, CNRS, INRAE, Centrale Lille, UMR 8207 - UMET - Unité Matériaux et Transformations, F-59000 Lille, France</footnote>

<footnote><sup>b</sup> Institut Universitaire de France, 1 rue Descartes, F-75005 Paris, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: pierre.hirel@univ-lille.fr (P. Hirel).</footnote>
the following. Negatively charged dislocations are sessile; positively charged dislocations are glissile so long as they keep their charge, but rapidly emit vacancies to become charge-neutral; and finally, charge-neutral dislocations are not dissociated but have a compact core, and they are sessile [14,15]. They concluded that the role of edge dislocations in the plastic deformation has been largely over-estimated in the past. Yet to account for experimental observations of dislocation loops at room temperature [6,11,16], and for multiplication of dislocations, both edge and screw components have to be mobile enough. If Klomp's conclusions that all variants of edge dislocations are virtually immobile held true, there would appear to be a shortage of slip systems to explain the ductility of STO at low temperature. In the following, we present a different charge-neutral dislocation core, that is dissociated and glissile, and restores the role of edge dislocations in plastic deformation.

Finally, as STO is heated above 1050 K (so-called Regime B), dislocations are known to lose their mobility, leading to brittle failure [2, 6]. In a previous study we showed that high temperatures could be responsible for a transition of edge dislocations into a climb-dissociated state which is sessile [17], however only positively charged dislocations were considered at that time. In the present study we investigate such core transformations for all charged and charge-neutral variants of edge dislocations.

# 1. Methods and models

## 1.1. Material modelling

Strontium titanate (SrTiO₃ or STO) is modelled in its cubic phase by means of a semi-empirical rigid ion potential that includes the Coulomb interaction, a short-range Morse function, and a repulsive term:

$$U(r_{ij}) = \frac{q_i q_j}{r_{ij}} + D_{ij} \left\{ \left[ 1 - \exp(a_{ij}(r_{ij} - r_0)) \right]^2 - 1 \right\} + \frac{C_{ij}}{r_{ij}^{12}} \quad (1)$$

where $q_i$ and $q_j$ are the electric charges of ions, $r_{ij}$ the distance between them, and $D_{ij}$, $a_{ij}$, $r_0$ and $C_{ij}$ are adjustable parameters. We use the parameters proposed by Pedone et al. [18], where $q_{Ti} = 2q_{Sr} = -2q_O = 2q$, where $q = 1.2|e|$. This potential was designed for broad usage and validated against the basic properties (lattice parameter, elastic constants) of strontium and titanium oxides, in addition to strontium titanate. We verified that different potentials yield similar results (see Suppl. Material). Coulomb interaction is evaluated by means of the particle-particle particle-mesh (pppm) algorithm with a relative accuracy of $10^{-6}$. Relaxation is performed by means of the conjugate-gradients (CG) algorithm, until total energy is converged within $10^{-9}$ eV/atom. All simulations are carried out with LAMMPS [19].

## 1.2. Simulation cell

A unit cell of STO is constructed with the crystal orientation $X = [110]$, $Y = [\bar{1}10]$, $Z = [001]$. It is then duplicated to form a $N \times N \times 1$ supercell, where $N$ was varied from 20 up to 100. We verified that results are sufficiently converged for $N = 80$, i.e. a cell of approximate size $440 \times 440 \times 4 \AA^3$, counting 64,000 ions (see Suppl. Material). Therefore, in the following results are presented for a size $N = 80$, unless specified otherwise. A dipole of edge dislocations is introduced by following a procedure similar to the one described by Marrocchelli and co-workers [12], as illustrated in Fig. 1. In the middle of the supercell, a ribbon of unit cells, of width [110], is removed, thus leaving a void region or gap in the material. One extremity is terminated with a plane of oxygen ions, therefore a negatively charged (\bar{1}10) surface, and the other by a plane of SrTiO₂ stoichiometry hence positively charged (Fig. 1a-c). The cell is compressed along the $X = [110]$ direction to compensate for the artificially negative pressure induced by the gap, and facilitate further relaxation. First, ions are allowed to relax only along the $X$ direction in order to close the gap that was formed. Then

they are allowed to relax in all directions. The cell geometry is then optimized to minimize stress in all directions. Relative energies are given in meV per angström to facilitate comparison between dislocations with different lengths.

Atomic systems are constructed with Atomsk [20]. Visualization is performed with VESTA [21] and OVITO [22]. To visualize defects, the centro-symmetry criterion [23] is computed using cations only (Sr and Ti) which form a bcc-like structure, so its value is naught for ions in perfect environment, and positive for ions in a defect (typically dislocations in the present study). The criterion is not computed for oxygen ions. Dissociation widths are estimated from the computation of disregistry functions, as detailed in Supplementary Material.

Results from atomistic simulations are notoriously sensitive to the interatomic potential employed, and to system size. We verified that the main results and conclusions reported in the present article remain unchanged by using a different interatomic potential, and that the system size yields converged results. More details about these points of comparison are presented in Supplementary Material.

## 1.3. Nudged elastic band calculations

To quantify mechanisms such as dislocation glide or core transformation, minimum energy pathways are computed by means of the nudged elastic band (NEB) method [24]. The computation uses 9 to 16 replicas, and a spring force of 1 eV/Å is applied between replicas to keep them equidistant. The calculation is considered converged when the maximum force on a replica falls below $10^{-2}$ eV/Å. Then the climbing image method [25] is applied so that a replica reaches the saddle point. The activation energy is then computed as the difference in total energy between the saddle point and the initial replica.

## 1.4. Charge density analysis

In the bulk material, electric charges of different ions are screened at short range. However it is not the case in defects, particularly in dislocations. We find that determining excess charges visually by looking at the positions of ions can be challenging and inaccurate, and subject to the appreciation of the person looking. Therefore, we rely on a more robust strategy, which we already employed to evaluate dislocation charges in MgSiO₃ perovskite [17]. This method is inspired by the Ewald summation method. Electric point charges of atoms are projected in the (001) plane normal to $Z$, and replaced by 2-D Gaussian functions that are evaluated over a regular grid. The charge density of an element of the grid is computed as the sum of contributions from Gaussian functions of all atoms:

$$\delta q(x, y) = \sum_{i=1}^{N} \frac{q_i}{\sigma \sqrt{2\pi}} \exp \left( - \frac{(x_i - x)^2 + (y_i - y)^2}{2\sigma} \right) \quad (2)$$

where the sum runs over all $N$ atoms in the system, $q_i$ is the charge of atom $i$, and $(x_i, y_i)$ its position in the (001) plane normal to $Z$. The standard deviation $\sigma = 3$ is chosen to yield a vanishing charge density in the bulk material. We verified that varying $\sigma$ from 1.5 to 4 has little influence on the results. Then, the charge of a dislocation is computed by integrating this charge density over an area surrounding the dislocation. We verified that integrating over the whole simulation cell yields a total charge close to zero, even if a very small residual charge may appear due to round-off errors.

# 2. Dislocation core structures

## 2.1. Dipole of charged dislocations

The construction method described above creates two surfaces with different stoichiometry: oxygen-rich on one side, and oxygen-poor (with SrTiO₂ stoichiometry) on the other side, as shown in Fig. 1(b) and (c). After relaxation, it leads to the formation of two edge dislocations
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Procedure to form a dipole of charged dislocations. Sr ions are represented as large spheres, Ti as medium spheres, and O as small spheres. TiO<sub>n</sub> polyhedra are represented, where n = 6 in the bulk, and n = 5 in some dislocation cores. Ions are coloured according to their central symmetry criterion (see methods), thus highlighting those belonging to dislocation cores. (a) In a supercell of SrTiO<sub>3</sub>, a ribbon of unit cells is removed. (b, c) Close-up views of the surfaces created by the cut. The top surface (b) has a stoichiometry SrTiO<sub>3</sub>, thus an excess of positive charge. The bottom surface (c) is a plane of oxygen ions and is negatively charged. (d, e) After relaxation, the gap closes and leads to the formation of two edge dislocations with opposite Burgers vectors, and different core structures. The APB is emphasized with a dashed line. (f) Computation of the charge density confirms that the top dislocation (d) carries a positive charge (in red), while the bottom dislocation (e) carries a negative charge (in blue). It also shows that the charge is localized on individual partial dislocations, the stacking fault between them being charge-neutral. Each partial carries a charge ±q/2 per length a<sub>0</sub> of dislocation line, or equivalently, a charge ±q per length of 2a<sub>0</sub> of dislocation line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** First scheme to neutralize dislocation charges (same colour code as Fig. 1). (a, b, c) After unit cells were removed from the initial system, an oxygen ion (per cell length along Z) is displaced from the bottom surface to the top one (red arrow). Then, the system is relaxed. (d, e) After complete relaxation, the system contains two equivalent dislocations with compact core structures, and opposite Burgers vectors ±[110]. (f) Charge analysis of the relaxed cell shows that both dislocations are charge-neutral.</figcaption></figure>

of opposite charges, as is well known from previous simulation studies [12,14,17]. Fig. 1(d) and (e) show the atomic configurations of the two dislocation cores, which are not equivalent. Their common features is to dissociate into two partial dislocations of Burgers vector $\frac{1}{2}[110]$, separated by a stacking fault known to be an anti-phase boundary (APB), as reported previously [8,26]. Their main difference is that in the positively-charged core, partial dislocations contain incomplete TiO<sub>5</sub> polyhedra due to missing oxygen ions (Fig. 1d). On the contrary, in the negatively-charged dislocation all octahedra are complete, thanks to excess oxygen ions (Fig. 1e). Computation of dis registries (see Supplementary Material) reveals dissociation widths of 33.5 Å and 31.25 Å for the positive and negative dislocations respectively. It also shows that the core of positively charged partials is more spread out in the glide plane, relative to negatively charged cores which are slightly more compact.

Fig. 1(f) shows the charge density analysis, revealing the charge on dislocations. Integration confirms that the bottom dislocation (in blue) has a total charge of -q i.e. the same charge as an oxygen ion, and the top dislocation (in red) a charge of +q i.e. the same charge as an oxygen vacancy. This analysis reveals another intriguing result, in that the charge is spread out. This was never reported before to the best of our knowledge. If we integrate only the region of a partial dislocation, we find that each partial carries a charge ±q/2 per line length. In addition, the APB between partials is effectively charge-neutral.

It may seem surprizing to find a charge that is a fraction of the charge of an ion. It means that on each partial dislocation, the charge of an oxygen ion is only partially screened by cations. In addition, it is important to realize that this is a lineic charge density *per length a*<sub>0</sub> of *dislocation line*, a point we will come back to later.

## 2.2. Full dislocation charge compensation

To balance electric charges, a first possible scheme is to displace one column of oxygen ions (of charge -q per length [001]) from the negative surface into the positive one, before relaxing the structure, as illustrated in Fig. 2. Such a procedure was already applied in SrTiO<sub>3</sub> to [100] dislocations by Marrocchelli et al. [12], and to [110] dislocations by Klomp et al. [14]. We also used a similar scheme in MgSiO<sub>3</sub> perovskite in our previous work [17].

Fig. 2(d) shows the atomic configuration of the dislocations after relaxation. The final energy of the system is lower by 6.71 eV compared to the cell containing charged dislocations, showing that charge compensation is very favourable. The charge-neutral dislocation is perfect and compact, i.e. non-dissociated, and two TiO<sub>5</sub> polyhedra share an oxygen ion (Fig. 2d, e). The details of its atomic arrangement differ from the one presented by Klomp et al. [14], who reported a more complex and shattered structure. Such difference may be due to the interatomic potential used, or to the specifics of relaxation procedure. Nevertheless, in both cases essential features are similar: the dislocation core is compact (not dissociated any longer), and it is sessile as will be shown later. Computation of the disregistry yields a core width of 7.8 Å, much more narrow than charged dislocations.

Since this dislocation is charge-neutral, one may expect electrostatic interactions to vanish, and elastic interactions to become dominant. Yet this compact structure appears to contradict Frank's criterion, and is also not consistent with the existence of a metastable stacking fault (APB). The issue seems to reside entirely in electrostatic effects. Indeed if the dislocation were to dissociate, it seems it would require formation of two partials of opposite electric charges to maintain neutrality. However oppositely charged partials would strongly attract each other
<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Second scheme to neutralize dislocation charges (same colour code as Fig. 1). (a, b) First, the simulation cell containing charged dissociated dislocations (as in Fig. 1d, e) is duplicated along the Z = [001] axis. Then, one oxygen ion is moved from each negative partial dislocation into a positive partial. (c, d) After complete relaxation, two equivalent dislocations are formed. They are dissociated, and each partial dislocation is charge-neutral. Along each partial dislocation line, one oxygen site out of two is vacant. (e) Charge analysis of the relaxed cell shows that both dislocations are charge-neutral. (f) Top view of a dissociated dislocation in (110), showing that each partial is formed of alternating positively and negatively charged segments. The system was duplicated 10 times along Z = [001] to obtain this image.</figcaption></figure>

and merge to form a compact core. From these considerations, we
understand that electrostatic effects, far from vanishing, bind this dis-
location into this compact configuration, and prevent its dissociation
completely.

## 2.3. Charge compensation of each partial

The previous charge-compensation scheme allows neutralizing the
total charge of the two dislocations in the cell. However the result-
ing compact core structure appears to contradict the well-established
dissociation of this type of dislocations. Our charge analysis revealed
that the electric charge was distributed evenly in the partial dislocation
cores, the APB between them being charge-neutral (Fig. 1f). As a result,
we can consider that in the simulation cell, there are four partial
dislocations, each carrying a charge ±q/2 per line length [001]. From
that perspective, moving an entire column of oxygen ions (of charge
−q per [001]) cannot neutralize the charge on partials, but can only
switch the sign of their charge between +q/2 and −q/2.

We propose another scheme, that allows each partial dislocation to be charge-neutral. First, we double the system size along Z = [001], resulting in a N × N × 2 supercell. By doing so, each partial dislocation now carries a charge ±q per cell length, which can also be expressed as ±q per length of [002] along the line. Then, out of the two oxygen ions along [002], one is displaced from each negative partial into its positive counterpart, as illustrated in Fig. 3(a, b). In other words, instead of displacing a full column of oxygen ions, we displace half a column from a negative partial dislocation, and another half column from the other negative partial. The total number of displaced oxygen ions is the same as in the previous scheme (i.e. a total of one oxygen per length a₀), but their distribution is different.

The resulting dislocation core structure is shown in Fig. 3(c) and (d). After relaxation, the dislocation remains dissociated into two partials separated by an APB. Along the dislocation line, oxygen sites are alternatively filled and vacant. As a result, each partial dislocation can be described as alternating negative and positive cores, as shown in Fig. 3(f), so that each partial is charge-neutral. Computation of the disregistry confirms the dissociation, and reveals a core width about 28.6 Å (see Supplementary Material). Charged dislocations were more spread out (about 32 Å), which we attribute to the fact that partial dislocations with the same charge repel each other, which increases the dissociation distance. On the contrary, after neutralizing the charge this electrostatic repulsion vanishes, so that the dissociation distance is controlled only by elastic effects.

Electrostatic effects being largely screened, elastic repulsion be-
tween partials of same Burgers vectors now dominate, and dislocations
are dissociated in their glide plane, separated by an APB. Note that
the periodicity of this dislocation line along [001] is not a₀ any longer,
but 2a₀. This allows for more degrees of freedom for relaxation. The

total energy of the cell is 1.18 eV lower than that of the cell containing
full neutral dislocations, or 7.89 eV lower than the cell containing
charged dislocations. To the best of our knowledge, this dissociated
charge-neutral core is the most favourable to be reported so far.

## 2.4. Neutral dislocation transformation

We found two neutral dislocation cores, namely the compact or
perfect dislocation (Fig. 2d), and the one dissociated in its glide plane
(Fig. 3c). To characterize possible transformation between those two
configurations, we perform NEB calculation. In this calculation, the ini-
tial configuration is the one containing two perfect neutral dislocations
(as in Fig. 2). The final configuration is obtained by shifting two rows of
oxygen ions along [110], one row by a₀/2 to the left, the other by a₀/2
to the right. After relaxation this configuration relaxes into a neutral,
dissociated core equivalent to the one in Fig. 3. Nine intermediate
configurations are constructed by linear interpolation between those
initial and final states, and then the NEB method is applied to optimize
the reaction path.

Fig. 4 presents the converged minimum energy path. As expected from previous results, the final configuration (dissociated) is lower in energy than the initial one (compact). The transformation requires crossing an energy barrier about 6 meV per angström of dislocation line. This calculation confirms that the compact core is metastable and hence can be relaxed in simulations, however the energy barrier is very small and would easily be crossed at finite temperature. Furthermore, the asymmetry of the barrier indicates that the reverse transformation, with an activation energy of 78 meV/Å, is much less likely to occur, so the dissociated dislocation would remain dissociated.

To verify if this mechanism can occur spontaneously during a
simulation, we performed MD simulation at various temperatures. The
initial system has a size 80 × 80 × 2 and contains a dipole of compact
neutral dislocations. At 1000 K the two dislocations dissociate after a
few ps. At 600 K it takes about 10 ps for at least one dislocation to
dissociate, and at 300 K it can take up to 100 ps. In any case it confirms
that the compact core is easily destabilized by thermal noise, and that
dissociation is much more favourable.

3. Dislocation motion

We submit the previous systems containing dislocation dipoles
(charged, compact neutral, dissociated neutral) to simple shear strain
ε<sub>xy</sub> to promote glide in (110), and monitor dislocation motion as well
as the resulting stress. All simulations are performed on systems of
size 80 × 80 × 2, meaning that the system size is doubled along Z =
[001] for charged and perfect neutral dislocations. We perform static
simulations, where the simulation cell is deformed by increments of
0.1%, and after each deformation increment, ions are allowed to relax
<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Minimum energy path for transforming the charge-neutral dislocation from a compact core (left) into a dissociated core (right). The energy of the latter configuration is used as reference of the energies, and all energies are given in meV per length of dislocation line. Same colour code as Fig. 1.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Athermal motion of the variants of [110] edge dislocations under applied shear strain in static simulations (in absence of temperature), obtained with NEB method (same colour code as Fig. 1). (a, b) The dissociated, positively charged dislocation is glissile, and both partials move simultaneously and smoothly through the lattice. (c, d) The dissociated, negatively charged dislocation is glissile, but only at finite temperature, and it moves abruptly from one site to the next. (e, f) The dissociated neutral dislocation is glissile, both partials move simultaneously and smoothly. (g, h) The perfect neutral dislocation is sessile, under stress it leads to formation of a crack in the material. (i) Minimum energy paths for the motion of negatively charged (blue dashed line with # symbols), positively charged (red line with +) and dissociated charge-neutral (black line with open circles) dislocations. Energies are given in meV per Å of dislocation line. Motion of dislocations are illustrated in Suppl. Videos SV1 to SV4.</figcaption></figure>

while the box geometry is fixed. This mimics quasi-static deformation with a zero strain rate, without accounting for temperature effects. Cell geometry was optimized previously, therefore pressure remains close to zero in these simulations.

### 3.1. Positively charged dislocation

We begin with the cell containing charged dislocations, one positive and the other negative (see Fig. 1). When imposing shear deformation under static conditions, only the positively charged dislocation moves,

while the negatively charged dislocation remains immobile. This allows us to investigate the motion of the dissociated, positively charged dislocation independently.

Fig. 5(a) and (b) show two snapshots of the positively charged dislocation moving under applied stress. This dislocation glides very easily, under a critical stress about 240 MPa. It moves as a rigid object, the two partial dislocations maintaining a constant dissociation distance. This behaviour is qualitatively consistent with previous results, however the critical stress (240 MPa) is significantly lower than the one reported in literature which is in the range between 1.2 and 1.5 GPa [8,14]. This
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** Procedure to form a dipole of charged dislocations dissociated in their climb plane (same colour code as Fig. 1). (a) In a supercell of $SrTiO_3$, two ribbons of width $dx = a_0\sqrt{2}/2$ are removed. One ribbon is shifted along the $Y = [\bar{1}10]$ direction with respect to the other. (b, c) Close-up views of the surfaces created by the cut, the top surfaces (b) have a stoichiometry $SrO$ or $TiO_2$, the bottom surfaces (c) are terminated with oxygen ions. (d, e) After relaxation, the gap closes and leads to the formation of climb-dissociated edge dislocations with opposite Burgers vectors, and different core structures. (f) Computation of the charge density confirms that the top dislocation (d) carries a positive charge (in red), while the bottom dislocation (e) carries a negative charge (in blue). It also shows that the charge is localized on individual partial dislocations, the stacking fault between them being charge-neutral. Each partial carries a charge $\pm q/2$ per length $a_0$ of dislocation line, or equivalently, a charge $\pm q$ per length of $2a_0$ of dislocation line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

large difference can be attributed to the use of different interatomic potentials.

Motion of the positively charged dislocation is triggered by incomplete $TiO_5$ polyhedra, in the vicinity of the leading partial dislocation, moving forward to connect to a complete $TiO_6$ octahedra, so that the pair forms an APB. On the trailing partial, a pair of polyhedra forming an APB disconnects, and allows the restoration of perfect crystal after the passage of the dislocation. As a result, the dislocation moves smoothly without leaving any kind of defect behind it. These elementary mechanisms are better displayed in Suppl. Video 1. The Peierls barrier associated with the motion of the positively charged dislocation is reported in Fig. 5i (red line with + symbols). We obtain an activation energy of 9.7 meV per angström of dislocation line (about 38 meV per unit cell length $a_0$). Increasing the load up to 70%, the positively charged dislocation still keeps moving by the same mechanisms over large distances.

## 3.2. Negatively charged dislocation

As explained before, when the system containing charged dislocations is sheared then the negatively charged dislocation remains immobile. This indicates that the negative dislocation is much less mobile than the positive one. To prevent motion of the positively charged dislocation, we freeze atoms surrounding it. Under these conditions, in principle only the negatively charged dislocation should be able to move.

However even in these conditions the negatively charged dislocation does not move. As the system is sheared and the applied stress builds up, at first the trailing partial moves while the leading partial remains immobile, thus narrowing the core. When the stress reaches about 2 GPa a crack forms from the leading partial and out of the (\bar{1}10) plane. This behaviour is animated in Suppl. Video 2. This means that in the absence of thermal activation, the negative dislocation is sessile. This is consistent with similar simulations performed by Klomp et al. [14,15].

We used the NEB method to characterize its motion from one Peierls valley to the next, and the minimum energy path is reported in Fig. 5i (blue dashed curve with # symbols). The activation energy is 24.3 meV/Å, i.e. about 2.5 times larger than for the positively charged dislocation. More importantly, the path around the saddle point is much steeper, indicating that this dislocation tends to move abruptly from one position to the next, rather than gliding smoothly along the plane.

The existence of an energy barrier means that negatively charged dislocations should move with the help of thermal activation. To verify this, we performed molecular dynamics simulations in the canonical (NVT) ensemble, where temperature is maintained constant by means

of a Nosé-Hover thermostat, shearing the box at a constant rate $\dot{\epsilon} = 10^8$ s$^{-1}$. The positively charged dislocation is maintained frozen to prevent its motion. In these conditions, the negatively charged dislocation does glide, although its motion appears very abrupt. This confirms that glide is very difficult for this type of dislocation.

## 3.3. Dissociated neutral dislocation

We now turn to the system containing two equivalent dissociated neutral dislocations (Fig. 3). Fig. 5(e) and (f) are two snapshots of such a dislocation moving (see also Suppl. Video 3). This dislocation is found to be mobile when the shear stress reaches about 70 MPa, which is about 3.5 times smaller than the stress required to move the positively charged dislocation. Ahead of the leading partial dislocation, an oxygen ion moves by $-1/2[\bar{1}10]$, thus making room for that partial to move forward. On the trailing partial, another oxygen ion moves by the same amount, thus restoring perfect crystal. Hence, this dislocation moves without leaving any kind of defect in its wake (Suppl. Video SV3).

NEB calculation reveals that the Peierls barrier is about 5.4 meV/Å (open circles in Fig. 5i). This is the lowest activation energy among all the dislocations studied here, meaning this dislocation is the most mobile through the lattice.

## 3.4. Compact neutral dislocation

Finally, we consider the system containing two compact neutral dislocations. Its behaviour under stress is illustrated in Fig. 5(g) and (h), and in Suppl. Video SV4. This dislocation remains immobile until the stress reaches about 1.2 GPa, and then a crack forms and propagates out of the (\bar{1}10) plane. The high stress level, and the fact that this dislocation is sessile, are consistent with the results of Klomp et al. [14].

Similarly to the negatively charged dislocation, we performed molecular dynamics to test if thermal motion would help. Here, no atom is frozen, so both dislocations have a chance to move. To avoid dissociation (like in Fig. 4) the cell size is reduced to $1a$ along $Z$. We tested temperatures of 100, 300, 600 and 1000 K, and at all temperatures both dislocations in the simulation remain completely sessile. Eventually cracks form from one dislocation, or from both. As a result it is not possible (nor is it relevant) to compute an activation energy for its motion through the lattice. Among all the (110) edge dislocations studied here, the compact neutral dislocation is the only one to remain sessile under all conditions considered.

# 4. Climb dissociation

Now we consider [110] edge dislocations dissociated in their climb plane, i.e. in the (110) plane normal to their Burgers vector.
<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Two different schemes to neutralize the climb-dissociated dislocations from Fig. 6 (same colour code as Fig. 1). (a) In the first scheme, a full row of oxygen ions is displaced from the negatively charged dislocation into the positively charged one (red arrow), yielding two equivalent dislocations (b) and (c), which are globally charge-neutral. (d) Computation of charge densities reveals that each dislocation is composed of one positively charged and one negatively charged partial dislocations. (e) In the second scheme, the system is first duplicated along the dislocation lines Z = [001], and then half oxygen columns are moved from each negative partial into a positive partial (red arrows), yielding two equivalent dislocations (f) and (g). (h) Computation of charge densities shows that each partial dislocation is charge-neutral.</figcaption></figure>

## 4.1. Core structures

Climb-dissociated configurations are constructed as follows. In a 80 × 80 × 1 supercell of $SrTiO_3$, two ribbons with a width of half a unit cell are removed next to each other. One ribbon is higher along the $Y = [\bar{1}10]$ direction than the other, so that the extremities of the cuts are L-shaped. Fig. 6a-c illustrates the procedure where the two ribbons are shifted by $4b$ with respect to one another. For the same reasons as previously, on the bottom side the two surfaces created by the cut are oxygen-rich, while on the top side the two surfaces are oxygen-poor. After relaxation, two climb-dissociated dislocations form, with opposite Burgers vectors and different core structures, as shown in Fig. 6(d, e).

Computation of charged densities confirms that the top dislocation (oxygen poor) is positively charged, while the bottom dislocation (oxygen rich) is negatively charged. In addition, it reveals that the charge is localized into individual partial dislocation cores, as evidenced in Fig. 6(f). Much like their glide counterparts, partial dislocations carry a charge of $\pm q/2$ per unit line length $a_0$. Note that despite their strong electrostatic repulsion, partial dislocations cannot move away from one another because they are bound by a stacking fault in the (110) climb plane, and their Burgers vectors are normal to the climb plane.

Similarly to glide-dissociated configurations presented above, there are two ways to compensate this net charge. The first scheme consists in displacing an entire oxygen column from the negative dislocation into the positive one, as illustrated in Fig. 7a. This procedure presents the same difficulties as for glide dislocations. Because an oxygen column has a charge $q$ per unit line length $a_0$, removing an entire column from a negative partial can only change its charge from $-q/2$ to $+q/2$; placing it into a positive partial can only change its charge from $+q/2$ to $-q/2$. Moving an oxygen column from any negative partial to any positive partial gives similar results. In the end, the two [110] dislocations are globally charge-neutral, but individual partial dislocations are not, as

evidenced by charge densities (Fig. 7d). Overall, we find that this procedure is not satisfactory, as it does not allow to neutralize individual partial dislocations, and charged partials act as strong electric dipoles.

The second scheme consists in doubling the system size along the $Z = [001]$ direction of dislocation lines, and then move half oxygen columns from each negative partial dislocation into its positive counterpart, as illustrated in Fig. 7e. After relaxation, one obtains two equivalent dislocations, as shown in Fig. 7(f, g). Along each partial dislocation, one oxygen site out of two is vacant. Computation of charge densities reveals that each individual partial dislocation is charge-neutral (Fig. 7h).

## 4.2. Core energies

We computed the total energies of the systems containing charged and charge-neutral dislocations. Fig. 8 shows the energies of the different configurations, as function of the height of dissociation in the climb plane. All energies are relative to the glide-dissociated neutral dislocation, taken arbitrarily as the origin of energies (black diamond at $h = 0$). The three points at $h = 0$ correspond to glide-dissociated configurations presented before, in order of increasing energy: dissociated charge-neutral dislocations (black diamond), compact neutral dislocations (blue circle), and charged dislocations (red crosses).

Points for a climb height $h > 0$ correspond to dislocations dissociated in their climb plane. For all dissociation heights, we find that the most favourable configuration is always the one where each partial dislocation is charge-neutral. The second most favourable is the one where dislocations are globally neutral, but contain charged partial dislocations, and the least favourable is the configuration containing charged dislocations. The minimum of energy is obtained for a dissociation height of 3 to $4b$, corresponding to the core structures presented in Fig. 7. These curves are consistent with the energies of climb configurations reported by Klomp et al. [14].
<figure><img src="image_10.png" /><figcaption>**Fig. 8.** Relative energies of supercells containing a dipole of charged dislocations (red crosses), a dipole of globally neutral dislocations (blue empty circles), and a dipole where each partial dislocation is charge-neutral (black empty diamonds), as function of the height of dissociation in the climb plane. The zero height corresponds to the dislocations being dissociated (or spread) in their respective glide planes. The glide-dissociated, charge-neutral core is taken as reference of energies.</figcaption></figure>

### 4.3. Glide to climb transition

Finally, we study the mechanisms of the transition from a glide-dissociated core into a climb-dissociated configuration. In a previous work, we already simulated this transition, and obtained an activation energy of 880 meV per Å of dislocation line [17]. However, at the time we only considered a positively charged dislocation, and the climb-dissociated configuration had a height of 2 unit cells. It can be argued that intermediate configurations may exist but have been overlooked in this previous work.

Fig. 9 shows the energy barriers and metastable configurations associated with the transition from glide to climb for both charge states. Let us begin with the positively charged dislocation (red dashed line). Initially dissociated in its glide plane (Fig. 9a), the dislocation must first reduce its dissociation distance (Fig. 9b). This comes at a relatively high cost about 600 meV/Å, which can be attributed to strong elastic and electrostatic repulsion between two positively charged partials. When the core is narrow enough, a Sr ion migrates from one (110) plane to the one below, as labelled in Fig. 9(b) and (c), thus allowing two TiO$_6$ octahedra to connect and form an APB in the vertical (110) plane, completing the climb dissociation (Fig. 9c). This mechanism is also animated in Suppl. Video 5. The energy barrier that we find here is comparable to the one from our previous work (880 meV/Å [17]), although a different interatomic potential was used, and the final configuration is slightly different. In any case, the climb-dissociated configuration is energetically more favourable than the glide one, meaning that the reverse transformation (from climb to glide) is much more unlikely to occur.

We performed a similar NEB calculation for the charge-neutral dislocation. Similarly, from its glide-dissociated state its has to first reduce its dissociation width (Fig. 9d–e). The main difference here is that it can relax into the neutral compact core, which is metastable as shown before. This first activation energy is about 78 meV/Å, which is an order of magnitude smaller than for the positively charged dislocation. While this compact core is sessile, it can easily transform back into a glissile state, as explained before (see Fig. 4). To truly climb, migration of a Sr ion is required, as labelled in Fig. 9(e) and (f).

This is associated with an activation energy of 9 meV/Å, effectively forming a vertical APB terminated by two partial dislocations. The resulting core structure (Fig. 9f) is lower in energy than the initial glissile core by 37 meV/Å. After that, further migration of oxygen ions, with an activation energy about 83 meV/Å, allows the connection of TiO$_6$ octahedra and minimizes the system's energy (Fig. 9g). Again this final climb-dissociated configuration is energetically more favourable than the glissile state by about 285 meV/Å, a value comparable to the difference found for a climb height $h = 1b$ above (310 meV/Å, see Fig. 8). The complete mechanism is animated in Suppl. Video 6. This confirms that climb dissociation is more favourable, even for small climb heights.

## 5. Discussion

### 5.1. Comparison with previous studies

In this work we report a new core structure for the edge dislocation, that was overlooked in previous works for two reasons: the lack of tools to evaluate the electric charge of dislocations, and boundary conditions that were too constraining and reduced the degrees of freedom.

In 2012 we reported the core structure of edge dislocations, and evaluated their Peierls barrier [8]. However at that time the electric charge of the dislocation was not investigated. In retrospect, it is now clear that the dislocation carried a positive electric charge, meaning that this study was incomplete. In 2023 Klomp et al. paid more attention to the electric charge of dislocations, which were estimated by counting oxygen ions in excess (or deficit) in the plane of cut [14]. However this crude estimation does not tell how the charge is distributed when the dislocation is dissociated: is it entirely located in the core of one partial dislocation, or inside the stacking fault, or is it spread out over the whole dissociation width? By replacing point charges with Gaussian functions we obtain a more reliable picture, showing that the charge is evenly distributed in the cores of the two partial dislocations, the stacking fault between them being charge-neutral (Fig. 1).
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Reaction energy paths for transition from glide-dissociated into climb-dissociated configurations obtained from NEB calculation (same colour code as Fig. 1). The initial glide-dissociated state is taken as reference of energy, and the system's total energy variations are normalized to dislocation line length (meV/Å). (a) Initially the positively charged dislocation is in its glide configuration, the APB is shown with a black dashed line. (b) The dissociation width is reduced. Ions displacements are small but the energy cost is high. (c) A Sr ion (labelled in b and c) migrates, forming a vertical APB. The dislocation is in a climb configuration. (d) Initially the charge-neutral dislocation is in its glide configuration. (e) The core adopts the compact (perfect) configuration. (f) The Sr ion (labelled in e and f) migrates, forming a vertical APB. The dislocation is in a climb configuration. (g) Oxygen ions migrate to restore octahedra in the APB, thus reducing energy. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

The second difference comes from boundary conditions. When simulating long straight dislocations, it is customary to use periodic boundary conditions along the dislocation line, with a period equal to one lattice constant. This is also what was done in SrTiO₃ by various groups, including ourselves [8], and by Klomp and co-authors [14,15]. Our present work shows that in the case of SrTiO₃, this severely restricts the possibilities to balance the charge of dislocations. The only possibility then is to move an entire column of oxygen ions from one dislocation to the other, which results in compact core structures (Fig. 2). We show that by doubling the period along the line, it is possible to construct a dislocation that is both charge-neutral and dissociated (Fig. 3), and that is lower in energy than all others reported in literature. This means that an even number of unit cells should always be used along the dislocation line, to form dissociated charge-neutral dislocations. On the contrary, imposing periodic boundary conditions with an odd number of unit cells effectively “kills” the dissociation, leaving only less favourable mechanisms to be activated, like glide of the positively charged dislocation.

In summary, our current results highlight the importance of assessing the actual charge on partial dislocations; and the critical importance of periodic boundary conditions, which can reduce too much the degrees of freedom and artificially prevent favourable mechanisms.

## 5.2. Glide of edge dislocations

In the context of plastic deformation of SrTiO₃ below 1000 K (Regime A), dislocations glide in {110} slip planes [1,2]. In such conditions, glide configurations are more relevant to account for plastic deformation. We begin by reviewing the properties of glide-dissociated dislocations.

To begin, the charge-neutral compact dislocation remains immobile at all temperatures studied. Even if by some unknown mechanism it happened to form in the lattice, it would easily dissociate to minimize its core energy (Fig. 4). Therefore, the role of the perfect neutral dislocation in plastic deformation can be completely ruled out.

The oxygen-rich, negatively charged dislocation (Fig. 1e) has two striking features. First, its core is enriched in oxygen with respect to the bulk. On the contrary, perovskite materials are notorious for the preferential formation of oxygen vacancies [27], which concentration can be tuned to enhance functional properties [28]. Moreover, such a negatively-charged dislocation would be a strong sink for oxygen vacancies, which would rapidly balance its charge. Second, we found that this dislocation moves with extreme difficulty, leaping from one Peierls valley to the next, with an energy barrier that is much higher than other variants (Fig. 5i). From these considerations, we conclude that both the formation and propagation of oxygen-rich, negatively
charged dislocations is very unlikely, and that they probably do not play any major role in the plasticity of SrTiO₃.

On the contrary, the positively-charged dislocation is reduced (Fig. 1d), which is more consistent with the existence of oxygen vacancies in the material. The fact that they have a good mobility (Fig. 5i) indicates that they can be efficient carriers of plasticity. In the long run, the charge of such dislocation segments could be compensated after external conditions stabilize and the sample is allowed to reach equilibrium, e.g. by vacancy diffusion, the formation of a Debye–Hückel cloud, and eventually the absorption or emission of vacancies, which would render the dislocation charge-neutral.

Finally the charge-neutral, dissociated dislocation has the lowest formation energy, and it is the most mobile among the dislocations investigated here (see Fig. 5i). The fact that it is charge-balanced means that it can remain as it is in the material, without requiring charge-compensation mechanisms, and is ready to glide if the material is deformed again.

An important outcome of our work is that two dislocation cores may exist and contribute to plastic deformation: the charge-neutral disassociated core, and the positively charged core. Both can glide relatively easily through the lattice, and both are compatible with the presence of oxygen vacancies. Since both cores are dissociated with similar widths, they may coexist along the same dislocation line. Long charged dislocation lines are probably unfavourable due to electrostatic effects, however a charge-neutral dislocation can interact with vacancies and contain short charged segments. Such charged segments may also balance their charge by emitting vacancies. Thus as it glides, a dislocation may interact with oxygen vacancies, some segments switching reversibly between charged and charge-neutral configurations, all while remaining glissile. The details of dislocation-vacancy interactions, and their role during plastic deformation (nucleation, multiplication, pinning), remain to be unveiled.

### 5.3. Diffusion and electrical switching

In 2006 Szot and co-workers demonstrated electrical switching in dislocation-percolated SrTiO₃ [29]. The authors showed that depending on oxygen partial pressure, pristine (undoped) SrTiO₃ crystals can switch reversibly from insulating to metallic behaviour. Metallic conduction was explained by electromigration of oxygen ions along dislocation lines, also known as pipe diffusion.

However pipe diffusion is still a matter of active research and ongoing debates. Atomic-scale simulations from different groups yield oxygen diffusion coefficients slower along dislocation lines than in bulk STO [12,30,31]. Either a different mechanism is involved in conductivity and electrical switching, or there exists a discrepancy between experiments and simulations.

The new core structure of (110) edge dislocations presented here may give a clue for enhancing diffusion. Indeed along charge-neutral dislocation cores, every other oxygen site is vacant, providing more degrees of freedom for ionic motion. This is also true in climb-dissociated dislocations, which may form in bicrystalline samples. Furthermore, edge dislocations in low-angle bicrystals may also be enriched in Ti and depleted in Sr ions, as suggested by high-resolution electron microscopy observations by Ikuhara and co-workers [4,32]. Thus when simulating diffusion, it is paramount to account for disorder and deviation from stoichiometry, as they may provide drastically different energy paths and enhance diffusion.

### 5.4. Ductile–brittle transition

Our simulations show that both charge-neutral and positively charged dislocations can transform in a climb-dissociated state (Fig. 9). First of all, we observe that the activation energy is much higher for positively charged dislocations, owing to the repulsion between positively charged partials that prevent their reunion. On the contrary, the

energy barrier for the charge-neutral dislocation is much smaller, and can be overcome at high temperature. Assuming a higher concentration and diffusivity of oxygen vacancies at high temperature, it is sensible to assume that charged segments would be rapidly neutralized, so that most edge segments would be charge-neutral. At 1000 K the factor $k_B T$ is about 86 meV, i.e. of the same order of magnitude as the energy barrier for glide to climb transition in charge-neutral dislocations (78 meV/Å, see Fig. 9). If we assume a thermally-activated mechanism with a rate proportional to $\exp(E_a/k_B T)$ based on simple transition-state theory, this activation energy is consistent with the ductile to brittle transition measured to occur around 1050 K experimentally [1, 2].

In addition, the glide to climb mechanisms described above are conservative, i.e. they do not involve the intervention of vacancies. Naturally, at high temperature the vacancy concentration is expected to be quite high. We suspect that non-conservative mechanisms, where the dislocation absorbs or emits vacancies, may facilitate the climb transition even further.

## Conclusion

We investigated different atomic configurations of the (110)(11̄10) edge dislocation in strontium titanate. When spread in the glide plane, four different core structures with very different properties can be obtained. Negatively charged dislocations, and a compact charge-neutral core already reported before, have very low mobility and probably do not contribute to plasticity. Positively charged dislocations have a good mobility, but their electric charge makes the formation of long segments unfavourable. We report a new charge-neutral, dissociated configuration that is energetically the most favourable and the most glissile reported so far. We argue that edge dislocations adopt preferentially this charge-neutral core. When interacting with vacancies, short segments can gain a positive charge while still remaining glissile. This charge can be balanced by emitting vacancies, restoring the charge-neutral glissile core. This scenario is consistent with electrostatic considerations because dislocations can be overall charge-neutral, but still accounts for the possible glide of charged dislocation segments.

Finally, we confirm that the dissociation of (110) dislocations in their climb plane is energetically more favourable than in the glide plane. If diffusion is efficient enough, e.g. at high temperature, it may promote transformation of edge dislocations into a climb-dissociated state which is sessile. To this date this mechanism remains the most plausible for explaining the loss of ductility of STO at high temperature.

## CRediT authorship contribution statement

**Pierre Hirel:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Patrick Cordier:** Writing – review & editing, Validation, Project administration, Funding acquisition. **Philippe Carrez:** Writing – review & editing, Validation, Methodology, Investigation, Data curation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 787198 – Time- Man. Computing resources were provided by the high performance computing center at University of Lille.
# Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.actamat.2024.120636.

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