What types of isometric transformations are we talking about here? One common case is a translation by a displacement vector R.

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1 1. Planar Defects Planar defects: orientation and types Crystalline films often contain internal, -D interfaces separatin two reions transformed with respect to one another, but with, otherwise, essentially, identical crystal structure. If the lenth scale of interest is lare enouh, the transformation relatin the reions is usually isometric, meanin that distances and anles between equivalent points within the reions are unchaned. The interface represents a defect with respect to a perfect crystal. Locally, the interface is confined to a distinct -D plane. This may also be the case lobally, so that the planar defect is truly confined to a plane over its entire extent. But there is a clear link to other types of -D, interfacial defects, which may spread over a non-planar, -D manifold, or membrane, so we will discuss these in the context of planar defect, too. When observin these defects in TEM, we can expect two limitin cases for the local orientation of the planar defect: we may be viewin the defect either 1) parallel to its plane, which present transverse variations in the crystal potential w.r.t. the incident beam, or ) perpendicular to its plane, in which case the variations are lonitudinal. What types of isometric transformations are we talkin about here? One common case is a translation by a displacement vector R. rrr Another possibility is a rotation or reflection, represented by a matrix M. r M r In the immediate vicinity of the planar defect, the potential could be quite complicated. But just a few unit cells or more away from the interface, the separate crystals should nearly resume their ideal, transformed structures. That s usually ood enouh to determine the type of defect that separates them. Translation: Influence on Fourier coeff s It is the chane in crystal potential that influences the wave function of an electron propaatin throuh the material that ives rise to any kind of imae contrast or diffraction sinature. Instead of r, electron diffraction mainly involves the structure function U r, or rather its Fourier components U. For a translation, we expect UrUrR Far from the defect, in the transformed reion, we expect the crystal to be essentially perfect, so we can represent it by a Fourier series. For a translation, the RLVs needed are unchaned:

2 Ur U e U e e Ue i rr ir ir ir The only effect is a phase shift of some of the Fourier components: ir i U Ue U e Notice that a component is only affected if R 0, that is, its RLV has a component alon R. We can think of a eneral -D boundary, which may have an oblique orientation w.r.t. the incident beam, across which the some particular Fourier component has chaned phase. Different types of planar defects can ive rise to various values of. But a particularly easy case to analyze is when for some reflection, so U U. Thouh we assumed the transformation was a translation, the influence of a reflection or rotation on certain RLVs may be the same. In particular, we can associate this sin flip of certain Fourier components with antiphase boundaries (APBs). If we are only concerned with effect observed under a two-beam condition, a phase shift or chane of sin of the active Fourier component may be all we need to analyze the diffraction and imain effects. Translation: influence on Bloch waves Let s say we have the situation above. We have solved for the Bloch waves above the defect and find: j j j ir ik r r C e e Now let s find the solution below the defect. It s oin to be the same as that above, except for the translation correspond to that of the crystal potential: j j r rr j j irr ik rr C e e j j j i k R ir ik r C e e e j j j ir ik r r C e e The factor j exp ik R is common to every component of every Bloch wave, so we can just drop it and write C C e C j j i R j i e As for the crystal potential, the only difference is a phase shift of some Fourier components of the Bloch waves: those for which R 0. Scatterin matrix (two-beam) I d like to predict the effects of a planar defect in oblique orientation in a two-beam condition, particularly the dark-field imae contrast. If the defect only causes a phase shift in the Fourier coefficient

3 3 of the active reflection, we can start with our dynamical theory from before. Since we already know how to do that problem, let s boil it down to its basics. The two-beam result can be summarized in matrices as: 1 1 i z z C C e C z 1 z C C i z 0 e This ives us the diffracted amplitudes at some depth z. We know that at the entrance surface: C 0 so we can find the excitation amplitudes usin C 0 Now we have 0z 1 0 C z C 0 z 0 Let s combine the product of matrices into one x matrix, called the scatterin matrix: P( z) C zc which ives 1 0z 00 P( z) z 0 Propaation across a planar defect (I) Below the planar defect (fault) we described previously, at a depth z t from the entrance surface, we have 1 0 z C zt z Interestinly, the eienvalues are the same as above the fault. It is only the Bloch-wave coefficients that have chaned 1 0 C C 0 e i As before, we have so 1 0 t C t 1 1 t C 0 t The scatterin matrix in this reion is P( zt) C ztc 1

4 4 which allows us to write 0 z 0 t P zt z t Propaation across a planar defect (II) The key inredient is the boundary conditions at the planar defect. It is usually sufficient to only require that continuity of the wave function 0 t 0 t t t Now the wave function below the fault is 0z 00 P ztp () t z 0 In fact, the boundary condition at the entrance surface of the foil also tells us that 0 01 and 0 0, so life just ets easier and easier. At the exit surface of the foil ( z T ) 0 T 1 P T tp () t T 0 Antiphase boundary APBs are probably the simplest type of planar defect, allowin us to test our understandin of how electrons propaate across such defects. Imaine a 1:1 alloy with random substitution of the two components on lattice sites. Now assume some orderin of these components, with the type of atom alternatin periodically in some direction ( a superlattice). Orderin has also occurred in some other part of the crystal, but with the two atoms exchaned in the orderin sequence a so-called antiphase domain. Where these two reions occur, we have an APB. Say the structure factor for a superlattice reflection on one side of the boundary is F fa fb. On the other side, it will be F fb fa F. Propaation across an APB (I) The eienvalue matrix can be written i 1w z e 0 iwz z e i 1w z 0 e in terms of w1 s. The Bloch-wave coefficients in a two-beam condition can be written

5 5 sin cos C cos sin where sin 1 1 w and cosw 1 w We will need to find the scatterin matrix 1 P( z) C zc Then, below the fault, we have C 1 0 sin cos C 0 1 cos sin Then we can find P( zt) C ztc 1 Propaation across an APB (II) Let s simplify some more by considerin the stron beam case ( s 0 ). Now 1, 1 and iz e 0 0 e z iz C ; C 1 1 Let s define at and bt. Now ia e 0 () t ia and 0 e First ib a e 0 ( T t) ib a 0 e ia e cos sin P( a i a t) ia e 1 1 isina cosa Next ib a e cos sin P( ba i ba T t) ib a e 1 1 isinb a cosb a Propaation across an APB (III) Now applyin the boundary condition at the entrance surface: 1 cos P( a t) 0 isina Puttin this all toether 1 cos sin cos cos P( ba i ba a ab T t) P( t) 0 isinba cosba isina isinab We finally have

6 6 0 T cos tt T isin t T Diffracted intensity across an APB We can look at this result as a variation in intensity of the diffracted beam as the electron propaates throuh the foil. Above the APB, the diffracted intensity is z sin z Below the APB, we have z sin t z The derivative di dz reverses at the APB. If the APB is at t T, the imae intensity for the APB is always dark, reardless of T. DF imaes of inclined APBs: influence of tilt If we set up a sample with APBs at the Bra condition and usin DF imain with a reflection that is sensitive to the phase, the APBs do, indeed, appear dark with respect to the surroundin material. Tiltin slihtly away from the Bra condition also ives some interestin imae contrast. First, the contrast reverses, so the APBs appear briht on a dark backround. A little more tilt causes the APB to show two briht frines on a dark backround.

7 7 Inclined APBs: two-beam analysis The eneral two-beam case takes a bit more work. From the form of the eienvalues 1, w w 1 we miht uess that somethin interestin happens when w 1 N, where N is an inteer. There is no need to work these out analytically every time; the problem is ideally suited for a computer. The calculations match what was observed experimentally. Stackin faults in fcc crystals Stackin faults involve an extra or missin plane in the stackin sequence of a crystal. We saw that the {111} planes play a special role in fcc materials. Say we label the normal stackin sequence as ABCABCA We can imaine removin a plane - C, for example - from the whole sample, or just a

8 8 portion of the sample. Then the stackin sequence in that reion is ABABCA This is an intrinsic stackin fault. If, instead, we insert an extra plane in some reion, ABCBABC, then we have an extrinsic stackin fault. Stackin fault eometry The linear boundaries of a stackin fault have some of the qualities of dislocations. But since we can t draw a Burer s circuit throuh a surroundin reion of perfect material, they are called partial dislocations. In [001]-oriented semiconductor films, the {111} planes are inclined from the interface. We can think the correspondin translation vector R for intrinsic and extrinsic stackin faults as bein in opposite directions, thouh the choice will depend on whether we consider R to be or Stackin fault imaes To view stackin faults in TEM DF imaes, we need to use a RLV for which R 0. If we use a reflection havin R 0, we may see that partial dislocations borderin the SF, but et no contrast from the SF itself. If we do have R 0, frines become appear alon the SF. The frine spacin depends on the foil thickness.

9 9 Twin boundaries in fcc crystals A common type of planar defect that arises in fcc crystals is twinnin. The stackin confiuration of <111> planes in fcc is closely linked to [0001] stackin in hcp, because the enery difference is usually fairly small. Sometime, so-called microtwins form that just span a few unit cells in width. The twin domains will share {111} planes with the matrix reions. The sinature of twins in fcc crystals is extra diffraction spots at and equivalent positions. These are especially evident on <110> zone axes, where two sets of {111} diffraction spots appear, which can each present transverse twin boundaries. Usin the extra spots for DF imain hihlihts domains of the twinned area, which may be embedded in matrix with a sinle, dominant orientation.

10 10 Oriin of 1/3{111} position spots Start with the matrix spots in a [110] diffraction pattern of a zincblende (fcc) crystal. In -D projection, a rotation by 180 about 111 is equivalent to a reflection across 11. Likewise, a rotation by 180 about 111 is equivalent to a reflection across 11. A little bookkeepin shows that the extra spots combine with the oriinal pattern at 1/3{111} positions. We don t immediately et all of the spots seen in the experimental pattern, thouh. Dynamical diffraction is needed to enerate the entire pattern.