On the Energy of < 100 > Coincidence Twist Boundaries in Transition Metal Oxides

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On the Energy of < 100 > Coincidence Twist Boundaries in Transition Metal Oxides

D. Wolf

To cite this version:

D. Wolf. On the Energy of < 100 > Coincidence Twist Boundaries in Transition Metal Oxides. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-142-C6-145. �10.1051/jphyscol:1980637�. �jpa-00220075�

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JOURNAL DE PHYSIQUE Colloque C6, supplément aun° 7, Tome 41, Juillet 1980, page C6-142

On the Energy of < 100 > Coincidence Twist Boundaries in Transition Metal Oxides (*)

D. Wolf

Materials Science Division Argonne National Laboratory Argonne, IL 60439, U.S A.

Résumé. — Les énergies de joints de torsion en coïncidence formés par rotation autour de l'axe < 100 > ont été calculées pour des oxydes stœchiométriques de métaux de transition de structure NaCl (comme NiO, MnO, et CoO). Pour cela, on a utilisé les potentiels ioniques de Catlow et al. Après relaxation, on trouve que les énergies des joints de torsion en coïncidence les plus classiques (désorientations de 36,87°, 22,62°, 28,07° et 16,26° cor- respondant aux inverses de densité de sites en coïncidence 1 = 5, 13, 17 et 25 respectivement) sont plus grandes que l'énergie d'une surface libre (100). Cela provient des interactions coulombiennes répulsives entre ions proches de « l'anti-coïncidence » (anion et anion, ou cation et cation) qui sont capables de vaincre les interactions cou-

lombiennes attractives entre ions en coïncidence (cation et anion). En conclusion, contrairement au cas des métaux, les structures des joints de torsion < 100 > dans les oxides et les halogénures alcalins pourraient ne pas être simple- ment déduites de la géométrie des coïncidences de réseaux. On n'a pas tenu compte de l'entropie.

Abstract. — The energies of coincidence twist boundaries formed by rotation about a < 100 > axis in stoichio- metric transition metal oxides crystallizing in the NaCl structure (such as NiO, MnO, and CoO) have been cal- culated using the ionic-type potentials of Catlow et al. for these materials. From a relaxation calculation it is found that the energies of the most prominent coincidence twist boundaries at 36.87, 22.62, 28.07, and 16.26°

(which correspond to inverse coincidence-site densities of I = 5,13,17, and 25, respectively) lie above the energy of the free [100] surface. The reason arises from the repulsive Coulomb interactions between ions in configurations close to anti-coincidence (anion on anion or cation on cation) which are capable of overcoming the attractive Coulomb interactions between the ions in coincidence configurations (cation ou anion and vice versa). It is con- cluded that in contrast to metals the structures of < 100 > twist boundaries in oxides and alkali halides may not simply be derived from the coincidence lattice geometry. Entropy terms are not considered.

1. Introduction. — Recent experimental evidence in support of the existence of high-angle coincidence twist boundaries in metal oxides [1] apparently similar in nature to their metallic counterparts [2] has greatly stimulated the interest in grain-boundary properties in oxides. Chaudhari and Charbnau [3] had predicted the existence of such boundaries for MgO on the basis of simple energy calculations. In contrast to metals, however, their energies were found not to scale with the density I '1 of coincidence sites.

The unrelaxed structure, for example, of < 100 >

coincidence twist boundaries on [100] planes not only shows the usual coincidences (in which an anion is positioned directly above a cation or vice versa) but also a number of almost anti-coincidences (in which anions or cations rest on top of each other; see figure 1). The consequent existence of both attractive and repulsive long-range Coulomb interactions between ions on opposite sides of the grain boundary suggests that ionic relaxation may play an important role in stabilizing such boundaries. For that reason, in this article relaxation has been taken into account

(*) Work supported by the U.S. Department of Energy.

in determining the properties of < 100 > coincidence twist boundaries in metal oxides with the NaCl structure. Since reliable interionic potentials are available [4], as our first example we have chosen NiO for our calculations.

2. Method of calculation. — 2.1 GRAIN BOUNDARY UNIT CELL. — The unit cell of the planar superlattice obtained by rotation of two [100] faces of NaCl-type crystals by 36.87° with respect to each other is shown in figure 1. xx, yx and x2, y2 are the Cartesian coor- dinate systems associated with the surfaces of two semi-infinite crystals 1 and 2 whose [100] faces have been brought in contact. xs and ys denote the coordi- nate axes of the superlattice thus obtained for an

inverse coincidence-site density of Z = 5. Other boundaries examined are those for Z = 13, 17, and 25 corresponding to rotation angles of 22.62°, 28.07°, and 16.26°, respectively.

A view parallel to the boundary [100] plane P (dashed line) is shown in figure 2. To later allow for relaxation of the ionic positions a variable number, n, of lattice planes between the two perfect crystals 1 and 2 is inserted. In figure 2, for example, n = 4.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980637

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O N THE ENERGY O F

<

¹⁰⁰⁾COINCIDENCE TWIST BOUNDARIES C6-143

P=5,<100>,0:36 87"

IN CRYSTAL 2 Me2+ IN CRYSTAL 2 t y2

o Me2+ IN CRYSTAL I 1 0

Fig. 1. - Superlattice formed by rotation of two [loo] surfaces of NaC1-type crystals by 0 = 36.87 degrees about the

<

¹⁰⁰⁾^axis.

The central square shows the ion distribution in the unit cell of the superlattice of this Z = 5 boundary.

Fig. 2. - Separation of the rigld ideal semi-infinite crystals 1 and 2 by a variable number of lattice planes in which the ions may relax.

The ion arrangement sketched is that for the C = 5 boundary the structure of which in the x-y plane is shown in figure 1. P indicates the boundary plane.

A starting configuration for our calculations is obtain- ed by chosing an initial separation d of the planes on the opposite sides of P. Note that the z axis is a four- fold rotation axis which substantially reduces the number of ions for which detailed calculations have to be performed.

2.2 INTERIONIC POTENTIAL. - The interionic potential of Catlow et al. [4] includes long-range Coulomb, and short-range Born-Mayer and dipole- dipole Van-der-Waals contributions. For simplicity, terms arising from the polarizability of the ions

which have been included in Catlow et al.'s potential via the shell model have been dropped in our calculations. This simplification reduces the calculated cohesive energy per molecule from U , = 41.01 eV to 40.94 eV. These potentials were found to reproduce t h e ideal-lattice and defect properties of NiO, MnO

COO, and FeO rather well [4].

2.3 ENERGY CALCULATION. - While the calculation of the short-range interaction energy of a given ion is straightforward, the summation over the long- range Coulomb interaction potential is rather involv- ed. The Coulomb energy of an ion in the unit cell may be decomposed as follows (see Fig. 2) :

where Uj-p denotes the interaction energy of ion j with the semi-infinite crystal

P

^(P⁼^1,Z). The third contribution to

UpU'

represents the interaction energy of j with all other ions in the boundary layer in which relaxation is considered. qj is the ionic charge of ion j while rj - ^rmdenotes an interionic vector.

According to Lennard-Jones and Dent [5], for the interaction of j with a [loo]-faced semicrystal

where x, y, z are the coordinates of ion j (in the unit cell) with respect to the surface of crystal 1 with an ion of charge 4, at the origin. As seen from figure 2, q 2 = - q l ; hence

where zo denotes the distance between crystals 1 and 2.

The self energy of ion j in a thin plate of finite thickness was determined by means of a modification of Ewald's three-dimensional method for the limit in which the unit cell is periodically extended in x and y only but not in z. Following Tosi's [6] terminology and description of that method one can show that for a thin plate of finite thickness

4j 4 m

q n ^-k ~ ( r ~ - r , ) - 1;. - ^rm A ^k, ⁿ

(unit cell)

where A = a2 (see Fig. 1) is the area of the unit cell in the x-y plane. For ( 100 ) twist boundaries

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C6-144 D. WOLF a/do =

@.

k, represents the set of two-dimensional reciprocal lattice vectors associated with the unit cell and its periodic extensions which are characterized by the vectors r,. Finally, the summation over n involves all ions in the unit cell while a denotes a parameter associated with the width of a two-dimensional Gaussian charge distribution. It is chosen so as to ensure rapid convergence.

2 . 4 METHOD OF RELAXATION. - Starting from equations (2)-(4) and the appropriate short-range potentials [4], the energy, forces, and second derivatives may be calculated for every ion by the method described above. The total energy q(rj) may be expanded about its unrelaxed position ry as follows :

Uj(rj) = uJ{rj") - ~ ~ ( ' 8 ) (rj - ry)

+

+ $ ( r j - ry) Wj(r?)(rj -rj"), (5) where F j denotes the force on ion j while W j is the symmetrical second-rank tensor of the second derivatives at the site of ion j. At equilibrium,

dL$(rj)/drj = 0

.

Therefore,

r j = rj"

+

WJT1(rj") FJ(rj") (6) may be used as the basis for the iterative minimization of the energy of the ions in the grain boundary.

Wy1(r7) determines the amount by which ion j is to be displaced in the direction of its force components in order to reduce its energy and force. This relaxation method invotving the second derivatives has the advantage of faster convergence over the simpler and more widely used gradient method in which the pro- portionality factor by which the ions are displaced in the direction of their forces is the same for all ions in a given relaxation step.

3. Results. - The energy of a twist boundary may be defined as

ugh ⁼

C

( u j - UO)IA ⁹

.i

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(unit cell)

where Uo is the cohesive energy per ion in the ideal crystal (see section 2 .2).

3.1 SURFACE ENERGY. - TO test the relaxation procedure described above, Ugb was calculated for a hypothetical C = 1 boundary which one obtains for 8 = 0. The energies thus obtained as a function of d (see Fig. 2) are shown in figure 3. For d % do, Ugh converges towards twice the surface energy, 2 o. For the unrelaxed surface we found 2 o = 2 140 erg/cm2 while after relaxation we obtained 2 o w 1 875 erg/cm2.

These results were obtained by considering the relaxation of the six lattice planes closest to the [loo] surface.

A larger number of layers did not change the results.

Fig. 3. - Grain-boundary energies as function of the separation d of nearest-neighboring layers at the boundary plane. Notice that within the accuracy of our calculations the four curves for Z = 5, 13, 17, and 25 coincide.

It was found that the cations in the surface layer relax by about 1

%

of do into the surface while the anions relax by a similar amount out of the surface. Both cations and anions on the second layer were observed to relax towards the free surface. The relaxation was found to extend as far as the fourth layer below the surface. The magnitude of the surface and relaxation energies as well as the characteristics and magnitude of the ionic relaxation are in general agreement with Tasker's surface energy calculations for many alkali halides [7].

3 . 2 GRAIN-BOUNDARY ENERGIES. - For all four coincidence orientations considered grain-boundary energies larger than twice the free-surface energy, 2 o, were obtained. They are shown as functions of d in figure 3. In spite of different partially relaxed starting configurations chosen, it was found impossible to reduce Ugb to values below the surface energy. In fact, for d > do it was not even possible to reduce U,, below its value for the unrelaxed configuration sketched in figure 2. However, by increasing d a lower-energy configuration was obtained in all cases. Note that the values of Ugb in figure 3 do not depend on C within the accuracy of our calculations.

4. Discussion. - The physical reason for the insta- bility of the ( 100 ) coincidence twist boundaries considered is probably closely related to the nature of the Coulomb interaction. With the exception of the ions at the coincidence sites, many ions change their total Coulomb interaction with the crystal on the other side of the boundary from being attractive (for 0 = 0) to being repulsive (for finite values of 0). For example, the unit cell for C = 5 (see Fig. 1) contains eight pairs of ions in almost anticoincidence configurations but only two pairs of ions in coincidence configurations.

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ON THE ENERGY OF ( 100 ) COINCIDENCE TWIST BOUNDARIES C6-145

In spite of the recent observation [I] of the C = 5 and C = 13 boundaries in MgO, the following obser- vations are thought to support the validity of the results presented :

1. For all values of C, U,, converges towards the same surface energy o which was determined inde- pendently.

2. The surface energy and structure as well as the magnitude of the surface relaxation agree rather well with the general findings for the alkali halides [7].

3. The unrelaxed energies and forces for the ions a t the grain boundaries do not depend on the number of lattice planes chosen in which the ions may relax.

This suggests that the Coulomb-energy calculation according to equations (1)-(4) is correct.

This leads us to conclude that the structures of ( 100 ) coincidence twist boundaries in oxides and alkali halides cannot simply be derived from the coincidence geometry. This is also suggested by some of the difficulties with which these boundaries have finally been manufactured. Instead, the structure of these

boundaries may involve (1) large displacements of ions from the coincidence geometry of the unit cell, (2) point defects in the grain boundary, or (3) serrated non-planar boundary planes on an atomic scale.

Unfortunately a comparison of our results with those of Chaudhari and Charbnau [3] is not possible since their cohesive energies per ion are higher in the crystal with the grain boundary than in a single crystal.

We note that their expression for the interaction energy of an ion above a [loo] surface with the semi- infinite rigid single crystal below does not agree with that of Lennard-Jones and Dent [5] (see equation (2)).

Acknowledgements. ^-I am grateful to N.L. Peter- son, M. Riihle, R. Benedek, and A. Rahman for stimulating discussions.

Note added in proof. ^-A similar calculation has recently been performed for MgO. Stable grain boundary configurations were found for this material.

See D. Wolf in Proc. Int. Symp. on Grain Boundary Phenomena, Chicago, April 1980.

DISCUSSION Question. - L. W . HOBBS.

Are the ion relaxations you calculate predominantly in the boundary or out of the boundary ?

Reply. ^-D . WOLF.

Since the boundaries were found not to be stable, the relaxations are predominantly away from the boundary. In the case of the surface energy calculation, the relaxations were also perpendicular to the surface. They agreed qualitatively with those of Tasker for the Alkali Halides.

Question. - L. SLIKIN.

It might be amusing to do this calculation on AgCl.

Here, the Ag+ has a very large quadrupolar deforma- bility, which would allow the ions to lock in to the non-cubic field, thereby stabilising the boundary.

Reply. ^-D. WOLF.

I believe that for ionic crystals with charges f 1 the situation is in general more in favour of the exis-

tence of such boundaries since the Coulomb contribution to the cohesive energy is by about a factor of four smaller than for charges

,

^2.Our calculations have, so far, not taken into consideration the pola- risability of the ions.

Comment. ^-A. B. LIDIARD.

I would like to comment that methods of energy minimisation such as you use rapidly become ineffi- cient as the number of variables increases. In these cases it is necessary to use one of the efficient methods developed by numerical analysts in recent years, eg the conjugate gradient method (Fletcher and Powell) or the variable metric methods (Fletcher).

The Harwell defect simulation programs (e.g. Devil, Hades, etc.) use such methods and allow one to model large systems without excessive computing require- ments (core space, cpu time). The same large systems cannot be studied by simple methods which minimise the energy successively with respect to the individual atomic position variables.

References

[I] SUN, C. P. and BALLUFFI, R. W., Scripta Met. (to be published). STONEHAM, A. M., Phil. Mag. 35 (1977) 177 and ibid.

[2] SCHOBER, T. and BALLUFFI, R. W., Phil. Mag. 21 (1970) 109 (to be published).

and ibid. 24 (1971) 165. [5] LENNARD-JONES, J. E. and DENT, B. M., Trans. Faraday Soc 24 (1928) 92.

[3] CHAUDHARI, P. and CHARBNAU, H., SurJ Sci. 31 (1972) 104. 161 Tost, M. P.. Solid State Phvs. 16 (1964) 1.

[4] CATLOW, C . R. A., MACKRODT, W. C., NORGETT, M. J. and

bl

^TASK%P. W., Phil. ~ a g . ' 3 9 (19'79) 119.