First-principles calculations on dislocations in MgO

ABSTRACT While ceramic materials are widely used in our society, their understanding of the plasticity is not fully understood. MgO is one of the prototypical ceramics, extensively investigated experimentally and theoretically. However, there is still controversy over whether edge or screw dislocations glide more easily. In this study, we directly model the atomic structures of the dislocation cores in MgO based on the first-principles calculations and estimate the Peierls stresses. Our results reveal that the screw dislocation on the primary slip system exhibits a smaller Peierls stress than the edge dislocation. The tendency is not consistent with metals, but rather with TiN, suggesting a characteristic inherent to rock-salt type materials.


Introduction
Ceramic materials have a wide variety of functions and are indispensable in our society.Despite the versatility of their properties, their inherent brittleness limits their applications.For several decades, including recent years, much effort has been devoted developing ceramics with plasticity [1][2][3][4][5][6], e.g. by introducing highly dense defects, controlling microstructures, and utilizing external fields.Ceramics with high plasticity are easily processed and withstand harsh environments, leading to a wider range of industrial applications.
Dislocations are one of the lattice defects in crystalline materials, playing a dominant role in material plasticity [7].When crystalline materials deform, the movement of dislocations results in smaller stresses than when the grains themselves move.Peierls stress is defined as the maximum stress required for a dislocation to move to an adjacent stable position at 0 K [8,9].It can be measured experimentally; however, reported values for the same material often show large deviations due to unintentional point defects in the test specimens or differences in experimental conditions [10][11][12][13][14][15].On the other hand, theoretical simulation is a suitable tool for estimating true Peierls stresses because it is basically calculated at 0 K and examines pristine materials [16][17][18].
There are two main computational approaches to calculate Peierls stresses and dislocation core structures.One is the Peierls-Nabarro (PN) model [8,9,19], which is still actively used despite its halfcentury history.In the model, the dislocation energies are divided into two contributions.The first is the elastic energy described by the elastic theory, while the second is the generalized stacking fault energies (GSFE), which are estimated using interatomic potentials or first-principles calculations [20][21][22][23][24][25].The PN model has the advantage of less computational costs, but it cannot fully incorporate atomic relaxation during dislocation glide, which may have a potential error in the Peierls stresses.An alternative approach which can overcome the disadvantage of the PN model is the direct modelling of the dislocation core structures [26][27][28][29][30][31][32][33][34][35][36][37].In principle, increasing the size of the supercells should systematically improve the accuracy, yet it also drastically increases computational costs.Hereafter, we refer to the direct modelling approach of the core structures as an atomistic simulation, as in Ref. [36], to distinguish it from the PN model approach.
In metals, dislocation core structures have been studied for several decades with atomistic simulations based on first-principles calculations.In particular, screw dislocations of body-centered-cubic (BCC) metals have been intensively investigated [26][27][28][29][30] because they have extraordinarily high Peierls stresses [38] and do not follow the Schmid law [39].Furthermore, the effects of doping around the dislocation cores on dislocation glide have also been investigated [32][33][34].However, there are limited studies on estimating the Peierls stresses of ceramics using atomistic simulations based on first-principles calculations, with the exception of TiN and Si [35][36][37].
Previous studies on the Peierls stresses in MgO are summarized in Table 1.They elucidated that the primary slip system of MgO at low temperature is 1/2<110>{110} [40], but it is still controversial over whether edge or screw dislocations glide more easily.Experimentally, the extrapolated critical resolved shear stress (CRSS) at 0 K, namely the Peierls stress, of MgO is approximately 150 MPa [12].However, Gorum et al. experimentally measured that the CRSS of MgO at low contamination was about 40 MPa and that at high contamination was about 180 MPa [41], which means point defects largely effect on Peierls stresses.The Peierls stresses of the 1/2<110>{110} edge and screw dislocations can be separately estimated using transmission electron microscopy (TEM); Singh et al. estimated 60MPa for the edge and 170 MPa for the screw dislocation [42].On the other hand, Moriyoshi et al. concluded that the Peierls stresses of both the edge and screw dislocations are about 70 MPa from the in-situ TEM observations [43].
Theoretically, Amanda et al. analyzed the temperature dependence of the screw dislocation by using the PN model and dislocation dynamics simulations [47,48].The results show that (i) the simulated temperature-behaviour based on the double kink mechanism was consistent with experiments, (ii) the Peierls stress of the screw dislocation (~150 MPa), was larger than that of the edge dislocation (~80 MPa), and (iii) the screw dislocation was dominant for the plastic deformation like BCC metals.These are consistent with the experimental CRSS [12] (Table 1) and the fact that the screw dislocations in the TEM images are longer than the edge dislocations [49].Still, the values are inconsistent with those of the low contamination sample and in-situ TEM observation [41,43], which may be due to the limit of the phenomenological model.
Therefore, in this study, we model dislocations in MgO by atomistic simulation and estimate their Peierls stresses from first principles.Consequently, we have found that the Peierls stress for the 1/2<110>{110} screw dislocation is about half of that for its edge dislocation counterpart.

Dislocation models and energies
To retain periodic boundary conditions, we need to introduce two equivalent dislocations with opposite Burgers vectors, namely a dislocation dipole, into the supercell.Figure 1 shows a schematic of our model containing a dislocation dipole.
Table 1.Previous reports on the calculated and experimental Peierls stresses for the four dislocations in MgO considered in this study.Our calculated values are also tabulated.Those marked with a single asterisk were not identified whether they were from the edge or screw.Values for TiN, derived from atomic models using first-principles calculations [36], are also listed.
The dislocation energies including the elastic energies depend on the supercell sizes.Therefore, we calculated the dislocation core energies as follows: where E tot; dipole and E tot; bulk represent the total energies of the supercell with the dislocation dipole and the bulk supercell, respectively.The '2' in the denominator represents the number of dislocations per supercell and l is the length of the dislocation line.E elas is the elastic energy and decomposed as follows: where E dipole is a dislocation dipole interaction in the supercell and E dipoleÀ img is that between periodic images.In this study, E dipole and E dipoleÀ img are estimated using the calculated elastic tensor and the anisotropic elastic theory [51,52], which is implemented in the babel program [53].E elas depends on the core radius of dislocations, which is set to 2b, where b is the magnitude of the Burgers vector.

Computational details
First-principles calculations were performed using the projector augmented-wave (PAW) method [54] as implemented in the Vienna ab initio simulation package (VASP) [55,56].The Perdew-Burke-Ernzerhof functional tuned for solids (PBEsol) [57] was used for exchange -correlation functional.PAW data sets with radial cutoffs of 1.06 and 0.80 Å for Mg and O, respectively, were employed.Mg 3s and O 2s and 2p orbitals were considered as valence electrons.Internal atomic positions were optimized with fixed lattice constants until the residual forces on the atoms were reduced to less than 0.005 eV/Å.The plane-wave cutoff energy was set to 400 eV for the optimization.
After structure optimization, we estimated the Peierls stresses and potentials using the nudged elastic band (NEB) method [39].When one dislocation moves, the quadrupole structure in Figure 1 deforms, which modifies the elastic energies in the supercell [27,51].To avoid this, we concurrently moved both dislocations in the same direction.We set the number of transition images as four.The transition states were optimized until the residual forces converged to less than 0.03 eV/Å.

Results
Elastic properties play a dominant role in dislocation energy and dynamics, especially in the long-range regime.Therefore, we initially confirmed that the PBEsol functional calculations are in excellent agreement with the experimental values [58] as shown in Table 2.
We then calculated the GSFE on the {100} and {110} planes (see Supplementary Figure S1).The predicted glide direction with the minimum energy is 1/2<110> for both planes, which is consistent with the minimum Burgers vector in MgO.Furthermore, the unstable energies are 1.96 J/m 2 for the {100} plane and 1.08 J/m 2 for the {110} plane, resulting in that the primary slip system is 1/2 < 110>{110} system.

Edge dislocations
Figures 2(a,b) show the atomic structures of the 1/2<110>{100} edge dislocation before and after relaxation, respectively.The atomic structures of the entire supercell are shown in the Supplementary Figure S2.The atoms outside the dislocation cores are only slightly displaced by the elastic field generated by the dislocations.On the other hand, the atoms optimized in the vicinity of the dislocation cores are relatively significantly displaced, up to 0.37 Å, by the structure optimization.This indicates the difficulty of elasticity theory in accurately describing the atomic structures of the dislocation cores.Both Mg and O atoms near the cores are found to be undercoordinated.The density of states (DOS) of the supercells with and without the dislocation shown in the Supplementary Figure S3(a,b) is found to be close to that of the unit cell.However, the dislocation core generates a small tail at the valence band top, as indicated by the black arrow in the figure.Thus, the edge dislocation can modify the electronic and optical properties.The atomic structures before and after relaxation of the 1/2<110>{110} edge dislocation are shown in Figures 2(c,d), respectively.Compared to the 1/2<110>{100} edge dislocation, its undercoordinated atoms are widely distributed on the (1 � 10) slip plane.This extended core structures have been observed experimentally by the scanning TEM [59].It is also noted that the atomic structure of the 1/2<110>{110} edge dislocation agrees well with that in TiN [36], whereas that of the 1/2<110>{100} edge dislocation in MgO differs from that in TiN [36] (see also Supplementary Figure S4).In TiN, we can clearly see extra two half planes in the 1/2<110>{100} edge dislocation, but we cannot see them in MgO due to the extension on the (001) plane of the dislocation.Their GSFE profiles in the <100> direction on {100} surface are shown in Supplementary Figure S5.Although the shapes of the profiles are similar to each other, TiN has higher GSFE than MgO, which may reduce the distribution of the core structure in TiN.Thus, the atomic structures of the same dislocation in the same rock-salt structure depend on, e.g. the elastic tensor and the nature of the chemical bonds.
We then calculated the Peierls stresses using NEB.The energies of the initial and final structures should be identical because the final structures are merely translated from the initial one by the Burgers vector.Indeed, the energy differences are found to be approximately less than 1 meV/Å at most (see Supplementary Figure S6), which attributes to the numerical error caused by the discrete Fourier transform.To reduce the errors, we symmetrize the profile by flipping it and taking the average of the profile before and after flipping it.The raw transition energy profiles are shown in Supplementary Figure S6.
The results are shown in Figures 2(e,f), Table 1, and Supplementary Table S1.The energy profiles and corresponding quantities are found to converge with a sufficiently large number of atoms in the simulation cells for the later discussion, especially about the order of the Peierls stresses.For both the 1/2<110>{100} and 1/2<110>{110} edge dislocations, it is found that the atoms moved only about 0.5 Å at most during dislocation glide; nevertheless, the dislocation positions were moved by 1/2[110], namely 2.97 Å.The 1/2<110>{110} edge dislocation has a smaller Peierls stress than the 1/2<110>{100} edge dislocation, which agrees with the experimental and theoretical finding that the main slip system of MgO is 1/2<110>{110} [40,46,48].
Figure 3 shows the elastic and core energies of the dislocations.The elastic energy of the 1/2<110>{110} edge dislocation differs from that of the 1/2<110>{100} edge dislocation.They should be the same in an isotopic medium.Thus, the energy difference is due to the elastic anisotropy.Also, the core energy of the 1/2<110>{110} edge dislocation is smaller than that of the 1/2<110>{100} edge dislocation, which is consistent with the fact that the main slip system is 1/2<110>{110} in MgO.It is noted that the core energies do not converge with respect to the supercell size in our calculations.One potential cause is electrostatic interactions between effective charges at the dislocation cores under periodic boundary conditions [60].Such effective charges, however, do not affect the Peierls stresses in this study because the relative positions of the core structures under periodic boundary conditions remain constant during the dislocation glide.

Screw dislocations
Figures 4(a,b) show atomic displacements of the 1/2<110> screw dislocations before and after relaxation, where the atomic displacements to [1 � 10] direction are plotted by the differential displacement (DD) method [61], where the lengths of the arrows are proportional to the relative shift of two neighboring atoms along the [1 � 10] direction (vertical direction of the paper) when inserting the dislocation in the bulk.The DD maps are overlapped on atomic structure of the bulk in Figure 4(a) or the optimized structures in Figure 4(b).The initial structure before relaxation in Figure 4(a) has a circuit core structure, while the optimized structure in Figure 4(b) has an core structure mainly on (110) plane.Furthermore, atoms in the vicinity of the dislocation core have displacements perpendicular to the dislocation line, namely the [110] direction, which is similar to the 1/2<111> screw dislocation of iron [62].
We then glided the dislocation positions on the (001) and ( 110) planes to estimate their Peierls stresses (see Figures 4(c,d), Table 1, and Supplementary Table S1).Interestingly, the two slip systems share the same atomic structures, but the Peierls stress in the {100} plane is more than a hundred times than that in the {110} plane.We reveal that the large difference is originated from the DD plot of the dislocation.Originally, the dislocation is distributed on the (110) plane as shown in Figure 4(b).For glide on the (001), the dislocation is distributed on the (001) plane during the glide (see the right panel labeled B in Figure 4(c)) and redistributed on the adjacent (110) plane (see the right panel labeled D in Figure 4(c)).On the other hand, for glide on the (110) plane, the dislocation distribution is kept on the (110) plane during the glide, which results in the small Peierls stress.

Comparison with the previous simulations and experiments
In the theoretical studies shown in Table 1, it was generally concluded that (i) the primary slip system is 1/2<110>{110} and (ii) the edge dislocations exhibit smaller Peierls stresses than the screw dislocations for both slip systems.While our calculation results agree with the conclusion (i), the Peierls stress of the edge dislocation is higher than that of the screw dislocation for the {110} plane.The discrepancy should result from the non-elastic energy in the PN mode estimated by the GSFE, which incorporates atomic relaxation only perpendicular to the slip plane.The core structure of the 1/2<110>{110} edge dislocation in Figure 2(f) was relaxed to the [110] and [1 � 10] directions.In such cases, the relaxation cannot be adequately described by the GSFE.For the 1/2<110>{110} screw dislocation, we can see the displacements perpendicular to the dislocation line in Figure 4(b).On the other hand, such displacements are not seen in the previous report [47].It is noteworthy that the magnitude relationship between the edge and screw dislocations aligns well with that observed in TiN, which was estimated using the direct modelling and first-principles calculations, similar to our study.
The TEM observations concluded that, for the primary slip system, the screw dislocations have a Peierls stress approximately three times higher than or equal to that of the edge dislocations [42,43], which is inconsistent with our calculations.In mechanical tests, it is well known that inevitable point defects significantly affect Peierls stresses, which may result in the inconsistency.Another possible cause is a Peach-Koehler force [7], which is an interaction between dislocations.Moreover, the calculated small Peierls stress of the screw dislocation and its compact core structure in Figure 4(d) may suggest that the dislocation multiplication is originated from the double-cross slip mechanism [43].

Conclusions
In this study, we modeled the atomic structures of edge and screw dislocation cores for the 1/2<110> {100} and {110} slip systems using first-principles calculations.For the edge dislocations, the 1/ 2<110>{100} dislocation exhibits a slightly higher Peierls stress compared to the 1/2<110>{110} dislocation.For the screw dislocations, the 1/2<110> {100} dislocation demonstrates a significantly higher Peierls stress than the 1/2<110>{110} dislocation.This notable difference in the screw dislocations is attributed to the spatial distribution of their cores.Additionally, within the 1/2<110>{110} slip system, the screw dislocation presents a lower Peierls stress than the edge dislocation.In metals, screw dislocations exhibit Peierls stresses several times higher than those of edge dislocations, while TiN shows a similar behavior to MgO.Hence, the relationship between edge and screw dislocations in the primary slip system could be a distinctive characteristic found in the ceramics with the rock-salt structure.

Figure 1 .
Figure 1.Schematic showing the dislocation dipole structure used in this study.The area shaded in light green represents a supercell, and the dotted areas show the repeated cells.The area shaded in dark blue marks a quadrupole structure.

Figure 2 .
Figure 2. Atomic core structures and energy profiles during glide of the edge dislocations.Core structures of the (a, b) 1/2<110>{100} and (c, d) 1/2<110>{110} (a, c) before and (b, d) after relaxation.The green and red circles are Mg and O atoms, respectively.Bonds with a length 1.15 times longer than Mg-O (2.10 Å) in the bulk are illustrated throughout this paper.(e, f) energy profiles of the 1/2<110>{100} and 1/2<110>{110} edge dislocations, respectively.The horizonal dotted lines represents zero energy.The initial and final structures are set at 0 and 1 in the x-axis, respectively.The numbers in the figures show the numbers of the atoms in the supercells.Black and white dislocation labels ('⊥') are initial and final positions of the dislocation centres.Letters of 'A' to 'D' in the panels of the energy profiles represent the transition states, whose core structures are illustrated in the right figures.

Figure 3 .
Figure 3. Dislocation elastic and core energies as a function of the distance between dislocation dipole.Filled marks are dislocation core energies while open marks are elastic energies.

Figure 4 .
Figure 4. Atomic core structures and energy profiles during glide of the screw dislocations.(a, b) core structures (a) before and (b) after relaxation.(c, d) energy profiles of the 1/2<110>{100} and 1/2<110>{110} screw dislocations, respectively.Black arrows are differential displacement maps (see text for details).The length of the arrows is normalized the largest differential displacement.Black and white squares are initial and final positions of the dislocation centres, respectively.The other details are the same as in Figure 2.

Table 2 .
Calculated elastic constants in GPa.