estructura: The Materials Science of Thin Films

Samer Elatrache V-17810600 estudiante de CRF

Crystalline Solids
Many solid materials possess an ordered internal crystal structure despite
external appearances that are not what we associate with the term crystalline-
Le., clear, transparent, faceted, etc. Actual crystal structures can be
imagined to arise from a three-dimensional array of points geometrically and
repetitively distributed in space such that each point has identical surroundings.
There are only 14 ways to arrange points in space having this property, and the
resulting point arrays are known as Bravais lattices. They are shown in Fig.
1-1 with lines intentionally drawn in to emphasize the symmetry of the lattice.
Only a single cell for each lattice is reproduced here, and the point array
actually stretches in an endlessly repetitive fashion in all directions. If an atom
or group of two or more atoms is now placed at each Bravais lattice point, a
physically real crystal structure emerges. Thus, if individual copper atoms
populated every point of a face-centered cubic (FCC) lattice whose cube edge
dimension, or so-called lattice parameter, were 3.615 A, the material known as metallic copper would be generated; and similarly for other types of lattices
and atoms.
The reader should realize that just as there are no lines in actual crystals,
there are no spheres. Each sphere in the Cu crystal structure represents the
atomic nucleus surrounded by a complement of 28 core electrons [i.e., (1s)'
(2s)* ( 2 ~()3~~ ()3~~ (3)d)~'O I and a portion of the free-electron gas contributed
by 4s electrons. Furthermore, these spheres must be imagined to touch in
certain crystallographic directions, and their packing is rather dense. In FCC
structures the atom spheres touch along the direction of the face diagonals,
i.e., [110], but not along the face edge directions, i.e., [lOO]. This means that
the planes containing the three face diagonals shown in Fig. 1-2a, i.e., the
(111) plane, are close-packed. On this plane the atoms touch each other in
much the same way as a racked set of billiard balls on a pool table. All other
planes in the FCC structure are less densely packed and thus contain fewer
atoms per unit area.
Placement of two identical silicon atoms at each FCC point would result in
the formation of the diamond cubic silicon structure (Fig. 1-2c), whereas the
rock-salt structure (Fig. 1-2b) is generated if sodium-chlorine groups were
substituted for each lattice point. In both cases the positions and orientation of
the two atoms in question must be preserved from point to point.
In order to quantitatively identify atomic positions as well as planes and
directions in crystals, simple concepts of coordinate geometry are utilized.
First, orthogonal axes are arbitrarily positioned with respect to a cubic lattice
(e.g., FCC) such that each point can now be identified by three coordinates
If the center of the coordinate axes is taken as x = 0, y = 0,
z = 0, or (0, 0, O), then the coordinates of other nearest equivalent cube comer
points are (1,0,0) (0, 1,O) (1,0,0), etc. In this framework the two Si atoms
referred to earlier, situated at the center of the coordinate axes, would occupy
the (0, 0,O) and (1/4,1/4,1/4) positions. Subsequent repetitions of this
oriented pair of atoms at each FCC lattice point generate the diamond cubic
structure in which each Si atom has four nearest neighbors arranged in a
tetrahedral configuration. Similarly, substitution of the motif (0, 0,O) Ga and
(1/4, 1/4, 1/4) As for each point of the FCC lattice would result in the zinc
blende GaAs crystal structure (Fig. 1-2d).
Specific crystal planes and directions are frequently noteworthy because
phenomena such as crystal growth, chemical reactivity, defect incorporation,
deformation, and assorted properties are not isotropic or the same on all planes
and in all directions. Therefore, it is important to be able to identify accurately
and distinguish crystallographic planes and directions. A simple recipe for
identifying a given plane in the cubic system is the following:
1. Determine the intercepts of the plane on the three crystal axes in number of
2. Take reciprocals of those numbers.
3. Reduce these reciprocals to smallest integers by clearing fractions.
The result is a triad of numbers known as the Miller indices for the plane in
question, i.e., (h, k, l ) . Several planes with identifying Miller indices are
indicated in Fig. 1-3. Note that a negative index is indicated above the integer
with a minus sign.
Crystallographic directions shown in Fig. 1-3 are determined by the components
of the vector connecting any two lattice points lying along the direction.
If the coordinates of these points are u1 , ul, w1 and u,, u,, w2, then the
components of the direction vector are u1 - u2, u1 - u , , w1 - w,. When
reduced to smallest integer numbers, they are placed within brackets and are
known as the Miller indices for the direction, Le., [hkl]. In this notation
the direction cosines for the given directions are h / d h 2 + k2 + 1 2 ,
k / d h 2 + k2 + 1 2 , l / d h 2 + k2 + 1'. Thus, the angle a between any two
directions [ h, , k, , 11] and [ h, , k, , /2] is given by the vector dot product
h,h, + k,k2 + / ] 1 2
Jh: + k: + 1: d h ; + ki + /;
cos Q = (1-1)
Two other useful relationships in the crystallography of cubic systems are
given without proof.
1. The Miller indices of the direction normal to the (hkl) plane are [ hkl].
2. The spacing between individual (hkl) planes is a = a , / d h 2 + k2 + 1 2 ,
where a, is the lattice parameter.
As an illustrative example, we shall calculate the angle between any two
neighboring tetrahedral bonds in the diamond cubic lattice. The bonds lie along
[ 11 11-type directions that are specifically taken here to be [i 'i 11 and [l 1 11.
Therefore, by Eq. 1-1,
1
3
cos a = ( I ) ( - 1) + I ( - 1) + (1)(1) = -- and
d 1 2 + 1, + l2 J( - 1)2 + (- 1)2 + l2
a = 109.5".
These two bond directions lie in a common (110)-type crystal plane. The
precise indices of this plane must be 010) or (1iO). This can be seen by noting
that the dot product between each bond vector and the vector normal to the
plane in which they lie must vanish.
We close this brief discussion with some experimental evidence in support of
the internal crystalline structure of solids. X-ray diffraction methods have very
convincingly supplied this evidence by exploiting the fact that the spacing
between atoms is comparable to the wavelength (A) of X-rays. This results in
easily detected emitted beams of high intensity along certain directions when
incident X-rays impinge at critical diffraction angles (8). Under these conditions
the well-known Bragg relation
nX = 2asin8 (1-2)
holds, where n is an integer.
In bulk solids large diffraction effects occur at many values of 8. In thin
films, however, very few atoms are present to scatter X-rays into the diffracted
beam when 8 is large. For this reason the intensities of the diffraction lines or
spots will be unacceptably small unless the incident beam strikes the film
surface at a near-glancing angle. This, in effect, makes the film look thicker.
Such X-ray techniques for examination of thin films have been developed and
will be discussed in Chapter 6. A drawback of thin films relative to bulk solids
is the long counting times required to generate enough signal for suitable
diffraction patterns. This thickness limitation in thin films is turned into great
advantage, however, in the transmission electron microscope. Here electrons
must penetrate through the material under observation, and this can occur only
in thin films or specially thinned specimens. The short wavelength of the
electrons employed enables high-resolution imaging of the lattice structure as
well as diffraction effects to be observed. As an example, consider the
remarkable electron micrograph of Fig. 1-4, showing atom positions in a thin
film of cobalt silicide grown with perfect crystalline registry (epitaxially) on a
silicon wafer. The silicide film- substrate was mechanically and chemically
thinned normal to the original film plane to make the cross section visible.
Such evidence should leave no doubt as to the internal crystalline nature of
solids.