Ceramic materials Carter Horton / fulltext 12 Dislocations

12.1 A QUICK REVIEW OF DISLOCATIONS

Line defects in a crystalline material are known as dislocations (unless they are disclinations, which will be ignored because they are much more difficult and not nearly as important in ceramics). In contrast to point defects, dislocations never exist in thermodynamic equilibrium because they have formation energies of 1 eV (or more) per atom along the line and there is no significant balancing entropy contribution as there is for point defects. They are almost always present in crystals because of how the crystal grew or because it was deformed. Therefore dislocations usually form due to nonequilibrium conditions, such as thermal and mechanical processing, or for thin films and single crystals, during growth. There are two special types of dislocation.

Edge dislocations

Screw dislocations

12.1a. The Burgers vector is perpendicular to the dislocation line, u. The Burgers vector will be opposite if the extra half plane is below the glide plane.

Screw dislocation: Successive atomic planes are connected to form the surface of a helix (or screw) around the dislocation line as shown in Figure 12.2 where the dislocation is perpendicular to the planes: like the core of a spiral, parking ramp. The Burgers vector is parallel to the dislocation line and can point up or down for a leftor right-handed screw.

Glide plane: This is the plane containing both the dislocation line and the Burgers vector.

The important result is that the strain energy of a dislocation is proportional to the square of the Burgers vector b and the shear modulus of the material.

σ

12 σε

ε

FIGURE 12.5 Elastic energy of a screw dislocation from a plot of stress versus strain.

We will assume linear isotropic elasticity throughout this discussion.

Compact Core versus Spread Core

If a dislocation dissociates into partial dislocations, the self-energy will be reduced providing the angle between the Burgers vectors is acute (they are parallel not antiparallel). As a simple demonstration, consider the possibility that 10 partial dislocations each with Burgers vector b/10 are formed. The strain energy is reduced by a factor of 10! Once formed, these dislocations will repel one another since they have parallel Burgers vectors. However, since the Burgers vectors of these partial dislocations are not

lattice vectors, a planar defect is created between each pair of partial dislocations. If the energy required to form these planar defects is greater than the reduction in the strain energy, then the dissociation will not occur. The bestknown example of dislocation dissociation is found in face-centered cubic (fcc) metals. The dissociation produces two so-called Shockley partial dislocations, which do not have parallel Burgers vectors.

In an fcc crystal, the special feature of this dislocation dissociation is that the planar defect formed between the two partial dislocations can be regarded as a mistake in the stacking of the close-packed layers as shown in Figure 12.6. The term stacking fault is then applied to all such planar defects and the energy per unit area is the stack- ing-fault energy (SFE or γ, but do not confuse this γ with strain). Of course, for the general concept, the planes do not need to be close-packed planes; you can stack any planes incorrectly!

When calculating the SFE, we do not equate the SFE to the energy required to create the fault; instead we con-

(A)

(B)

FIGURE 12.6 The intrinsic stacking fault in an fcc crystal.

sider the force balance. The two partial dislocations will try to repel each other (bP1 · bP2 > 0), but they cannot separate too much because of the SF. Although we have deduced this dissociation for fcc metals using an energy argument, it is also important for the behavior of the dislocations. In fcc metals, these partial dislocations allow the atoms to move over “valleys” instead of over “hills” as the dislocation glides, and they restrict glide to the plane of the stacking fault. In Cu and Si, dislocations are widely ( 4 nm) dissociated, but the same dislocation in Al has a rather narrow core.

If the dislocation changes its line direction for a short segment, this segment is called a kink if it lies in the glide plane or a jog if it causes the dislocation to step out of its glide plane.

In general, a dislocation will itself contain many such defects along its length.

12.2 SUMMARY OF DISLOCATION PROPERTIES

We have not reviewed all the properties of dislocations, but have concentrated on those you should know when considering dislocations in ceramics.

Dislocations cannot end inside the crystal lattice.

We almost always assume that strains are so small that linear elasticity is a good approximation; so, Hooke’s law holds.

The displacement field for any dislocation falls off as r −1 as we move away from the dislocation.

The strain energy (or self energy) of a dislocation actu-

ally depends on the character of the dislocation, but setting E = αGb2 is a good estimate, where α is 0.5.

Parallel dislocations repel one another if the angle between their Burgers vector is less than 90°. (Be careful with u.)

A dislocation can always lower its strain energy by spreading its core or dissociating. Whether this will lower the total energy depends on the energy required to form the distorted region between the “partial dislocations.” If dissociation occurs (forming identifiable partial dislocations) then this region is called a stacking fault.

Dislocations glide by the movement of kinks and climb by the movement of jogs. Since climb requires changing the number of point defects (reacting or absorbing them), we call it nonconservative motion.

12.3 OBSERVATION OF DISLOCATIONS

We can divide the techniques used to “see” dislocations into two categories:

1.Direct

2.Indirect

Direct techniques allow us to see the arrangement of atoms around the dislocation. The most useful direct technique is high-resolution transmission electron microscopy (HRTEM). The resolution of commercial instruments allows direct observation of columns of atoms and edge dislocations can be identified as terminating planes of atoms as shown in Figure 12.7. These images look very much like the schematics we draw to illustrate edge dislocations, and the Burgers vectors can be determined directly from the image using the Burgers circuit construction (unless there is a screw component present).

Indirect techniques rely on the fact that dislocations create a strain field and are regions of high energy. The most widely used of the indirect methods is again TEM. We will show you several examples of TEM images of dislocations in ceramics in later parts of this chapter. Figure 12.8 shows how planes near a dislocation may be bent into an orientation where they are strongly diffracting even when the rest of the crystal is not. The intensity of the direct beam will be reduced (and that of the diffracted beam increased) and the dislocation will appear as a dark line in the bright-field image. Images of dislocations are often quite difficult to interpret and the position of the dislocation in the TEM image will generally not correspond exactly to its actual position in the crystal. However, this approach is extremely useful in answering many of

FIGURE 12.7 HRTEM of the core of a dislocation in Al2O3.

FIGURE 12.8 Diffraction contrast from dislocations in Al2O3.

the questions that arise about the nature of dislocations in

amaterial.

Is the dislocation interacting with other dislocations, or with other lattice defects?

Is the dislocation jogged, kinked, or straight?

What is the density of dislocations in the sample?

Has the dislocation adopted some special configuration, such as a helix?

By tilting our specimen in the TEM to a condition in which the dislocations are invisible (or at least appear faint in the image), if we know the diffracting conditions (specifically the diffraction vector g), then we can obtain b using the invisibility criterion (generally needing three independent g vectors).

Another indirect (but lower resolution) approach is the etch-pit method. In this technique the surface of the sample is polished and immersed in a suitable chemical etchant. The rate of removal of material around the dislocation (where the crystal is strained) is usually more rapid than in the surrounding crystal and pits are formed where the dislocation line intersects the surface. The clearest demonstration that etch pits are associated with dislocations is shown for Si in Figure 12.9a where the dislocations can be seen in cross section due to absorption of IR illumination by the Cu decorating the dislocation; today, this task might be more easily accomplished using a focused ionbeam (FIB) instrument. Figure 12.9b shows a plan-view image of etch pits in a single crystal of Nd-doped YAG. The etch pits were revealed by immersing the polished crystal in concentrated phosphoric acid at 250°C for 8 minutes.

The etch pit method is usually best when dislocation densities are low (<104 mm−2). If the dislocation densities are too high the etch pits overlap each other and it is very difficult to resolve their shape and count densities.

(A)

(B)

FIGURE 12.9 Etch pits at dislocations emerging at a surface in Si.

(a)The cross-sectional view in Si; (b) the plan view in YAG.

12.4DISLOCATIONS IN CERAMICS

In discussing dislocations in ceramic materials, the principle is always the same. We deduce the possible Burgers vectors first, then the glide planes. We will begin by asking if there is anything special about dislocations in ceramics; we can preempt the answer by saying yes, as usual, the bonding and charge can add their own effects. The second best-known special feature of dislocations in ceramics is actually that the unit cell of such materials is usually larger than for the simple metals, so we will see similarities to other materials such as the intermetallics that also have large unit cells.

The structure of the dislocation core in ceramics depends on three factors

1.Charge of the ions

2.Size of ions

3.Presence of directional bonds

Perhaps the more important question is: why learn about dislocations in ceramics when ceramics do not deform plastically as easily as metals? This question then leads us to question why this statement is true. Is it always true? Can we change anything? Modern materials often involve interfaces. Interfaces are directly related to dislocations. The growth of thin films often involves dislocations. Dislocations also play an important role in radiation damage of ceramics. So there are many reasons for understanding dislocations in ceramics.

12.5 STRUCTURE OF THE CORE

Not much is really known about the core of dislocations in ceramics. For the examples we will show, you should remember that we have usually chosen one Burgers vector (usually the most important one), one line direction, and thus one glide plane. Furthermore, we usually draw the edge dislocation because it is easiest to draw, not because it is the most important. In this section, we will assume that the dislocation core is compact. The examples are chosen to illustrate particular features.

NaCl: it is relatively simple and illustrates the effect of charge.

Si: it illustrates the effect of directional (covalent) bonding.

Al2O3 and olivine: they are noncubic materials.

Although NaCl and MgO structures are both based on the cubic-F Bravais lattice, like Cu, there is no evidence for any dislocation dissociation. There are detailed atomistic calculations that confirm that the compact core is preferred. A schematic diagram of such a dislocation viewed along the [001] direction is shown in Figure 12.10. All the ions seen here lie in the same (001) plane. If we remove this plane of atoms the structure looks the same, but all the ions are reversed in sign. So charge is balanced as long as there are no jogs or kinks on the dislocation.

The glide plane for dislocations in NaCl and MgO is {110} rather than the {111} you might expect for fcc. You may read that the glide plane is {110} because this plane is electrically neutral and motion on this plane avoids charged layers gliding over one another. However, ErAs, an exotic semimetal (no ionic charge), with the NaCl structure, has the same glide system. PbS, a semiconductor with this structure, shows glide occurring on {001} planes and in TiC dislocations glide occurs on {111} planes. The real reason for the {110} glide plane is still being studied. The suspicion is that the core actually does spread on different planes, but that still does not tell us why the spreading depends on the material.

The simplest covalently bonded ceramics are Si and Ge. The covalent bond formed by two atoms sharing electrons is a localized and directional bond. Cubic ZnS has the same structure, but the Si–Si basis is replaced by Zn–

qA = + 18 e qB = – 18 e qC = – 18 e qD = – 18 e

qAÕ = – 18 e qBÕ = + 18 e qCÕ = + 18 e qDÕ = + 18 e

S, although the bond still has a large covalent component. This feature is important in determining the characteristics of dislocations in covalent materials. Dislocations in these materials tend to be immobile at low temperatures. Extensive slip occurs only at elevated temperatures.

Of the many covalent crystals, the diamond-cubic and c-ZnS structures are among the simplest and most widely studied. Since the crystal lattice is fcc, perfect dislocations

dislocations in fcc metals, dislocations in Si, Ge, and c-

ZnS glide on {111} planes. Figure 12.11a shows a ¯ (110)

projection of c-ZnS; you can recognize the {111} planes (also shown by the projected tetrahedron). We make a dislocation by cutting out half a plane, as shown by the box labeled 1564. Then we merge the atom at P and Q; since these are on different levels (one is a large circle and the other is small), the dislocation will have a component of b into the page: it is a 60° perfect dislocation. However, we could have removed the slice of material 1234 and joined atoms R and S to make exactly the same b. Therefore, there are two possible {111} slip planes, type I and type II, and two possible dislocation cores. [These slip planes are usually called the glide (I) and shuffle (II) planes, which can be confusing since the dislocation can glide on both!] The 60° dislocation is shown with its extra plane ending at a type II plane. Imagine removing material 1234: there is then one unpaired bond per atom along the dislocation core as shown in Figure 12.11b. This defect is called a dangling bond. This type of dislocation does exist as illustrated in Figure 12.11c. Here, the dislocation alternates between the compact core and a dissociated structure. The explanation for this change is that the dislocation

steps from a type I plane to a type II plane: it dissociates on one but not on the other because the SFE is different on the two planes. The step from the shuffle plane to the glide plane is a special jog. The dangling bonds can then reconstruct; now the bonds are distorted but not dangling! However, this dislocation cannot move without rebreaking the reconstructed bonds so the dislocation is sessile.

If we left the half-plane 1234 where it is but removed the other half, we would create a dislocation with the opposite b, but instead of the last atom being R(S) it would be T(U): Zn instead of S! The core of the dislocation is fundamentally different: it is still a shuffle dislocation. These two dislocations are fundamentally different because the zinc blende structure does not have a center of symmetry. Similar considerations will hold for materials such as AIN and GaN, which also lack a center of symmetry but have the wurtzite structure crystal structure. The stacking fault in the diamond-cubic structure is described as AbBbCcBbCcAaBb: the pair of planes Aa behaves just as if it were one fcc plane in this case. Two possible SFs are shown in Figure 12.11d.

The thoroughly studied metals are either cubic or they have the closely related hcp structure. Many ceramic materials are neither cubic nor hcp. Olivine and sapphire both have oxygen sublattices that can be thought of as distorted hexagonal close packed (hcp), but the distribution of cations makes them very different.

Olivine. In the olivine group of minerals (important orthorhombic silicates), the energies of the [100], [010], and [001] dislocations are all different. In fact, the energy of the [010] dislocation will be much greater since it has a much larger Burgers vector.

B

A

C

I

A

I

B

A

For forsterite (Mg2SiO4): a = 0.475 nm, b = 1.020 nm, and c = 0.598 nm.

For fayalite (Fe2SiO4): a = 0.482 nm, b = 1.048 nm, and c = 0.609 nm.

For monticellite (CaMgSiO4): a = 0.4815 nm, b = 1.108 nm, and c = 0.637 nm.

The added complication is that olivine is not an isotropic material so using μ and ν (which automatically implies elastic isotropy) would be a simplification.

Sapphire. Al2O3 is also very anisotropic although we can use pseudoisotropic values of μ and ν when dislocations are confined to lie on the basal plane. The most common perfect dislocations do have the shortest

other perfect dislocations have been reported including [0001]. Even when they have the shortest b, they may

¯

glide (if they do at all) on other planes such as {1100}. Examples of dislocations in Al2O3 are shown in Figure 12.13.