Page 64 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

OVERVIEW 45

referred to as crystallites or grains and are separated from each other by grain boundaries.
If the individual crystallites are reduced in size to the point where they approach the size
of a unit cell, periodicity is lost and the material is called amorphous or glassy.
The geometric shape of a unit cell is a three-dimensional parallel-piped structure and
contains one or a few of the same 3 atom in simple crystals such as copper, sodium, or
silver but may contain thousands of atoms in complex organic crystals. The length of an
edge of the unit cell is called the lattice constant. The variety of crystal structures can
be defined by arranging atoms systematically about a regular or periodic arrangement of
points in space called a space lattice. A lattice is defined by three fundamental translational
vectors a, b, and c, so that the arrangement of atoms in a crystal of infinite extent looks
identical when observed from any point that is displaced a distance r from an origin, as
viewed from the point R, where

R — r + n 1a + n 2b + n 3C (3.2)

n 1, n 2, and n 3 are integers. This is schematically shown in Figure 3.10. The points
described by the position vector R constitute the space lattice.
Parallel planes of atoms in a crystal are identified by a set of numbers called Miller
indices. These numbers can be obtained in the following way: choose the origin of the
coordinate system x, y, z to coincide with a lattice point in one of these parallel planes.
Find the intercepts of the next parallel plane on the x-, y-, and z-axes as x 1, y 1, and
Zi, respectively. Now take the reciprocals of these numbers and multiply by a common
factor so as to obtain the three lowest integers h, k, and l; these integers are called Miller
indices and are normally written within parentheses (h, k, /). An example of this method
for plane identification is shown in Figure 3.11. If one of the intercepts happens to be a
negative number, an overbar is added to the corresponding Miller index to indicate that
the plane intercepts with the negative axis.
A crystallographic direction in a crystal is denoted by a square-bracket notation [h, k, l].
The numbers h, k, and / correspond, respectively, to the x, y, and z components of a
vector that defines a particular direction. Again these numbers h, k, and / are the smallest
integers, the ratios for which are the same as those of the vector length ratios. An overbar
on any of the integers denotes a negative vector component on the associated axis. For
example, the direction of the negative x-axis is [1,0, 0], whereas the positive .x-axis is
denoted by the direction of [1,0,0].

Figure 3.10 Representation of a two-dimensional crystal lattice in terms of fundamental transla-
tion vectors. A third unit cell vector c provides the third dimension (not shown)
3
The number depends on whether the unit cell is SC, BCC, or FCC in a cubic crystal lattice (see later).