Download Miller indices are a notation used to describe lattice patterns in a

Navid Fanaeian
Christian Schneider
ENEE 416: IC Fabrication
Group Activity 1: Miller Indices
September 18, 2007
Miller indices are a notation used to describe lattice patterns in a crystal. Lattice patterns
tell the orientation of surface planes within a crystal structure. There are three numbers,
in vector notation, that form a miller index. These numbers are found by using the
intercepts on three dimensional axes and can be used to uniquely identify a plane or
surface in a crystal. This is useful in the process of creating a silicon wafer.1
During the Czochralski Method of creating a silicon ingot a seed is dipped into molten
silicon. This seed has a specific orientation characterized by the Miller index. This seed
propagates its orientation to the ingot, making the structure similar throughout.2
Knowing the Miller indices of the ingot allows for a more even cut of the ingot into
wafers. Cutting along a plane with lasers will reduce the variations in the wafer.
The Miller index can also tell the density of nodes, or atoms in a crystal. Optical
properties, such as velocity of light behaving differently over densities of planes that
are indicated by the Miller index, can be known ahead of time by knowing the Miller
index value. Other properties affected are the absorption of light, surface tensions, and
the dislocation of atoms within the crystal structure.3 In order to understand the impact of
miller indices, we must first understand how to calculate it.
For any given plane of atoms in a crystal, the Miller indices for that given plane are found using
the following procedure:
1. Set up coordinate axes along the edges of the unit cell.
2. Note where the plane of interest intersects the axes.
3. Normalize the intercepts by dividing each of them by the number of unit cell lengths
along each coordinate axis and record the numbers in the order x, y, z.
4. Take the inverse of each of the normalized intercept values.
5. Multiply the inverted intercepts by the constant that would result in the smallest possible
set of whole numbers.
6. Insert the whole numbers in parentheses ( ).
The procedure to find Miller indices is best illustrated using an example. Picture a crystal unit cell
with sides that are 5 units in length. Consider the plane containing the triangle that intersects the
x, y, and z axes at 5, 10, and 15 units, respectively. To characterize this plane, we normalize the
1
Brigham Young University Integrated Microfabrication Lab, Crystal Planes in Semiconductors, Miller
Indices, http://www.ee.byu.edu/cleanroom/EW_orientation.phtml (as of Sep. 17, 2007, 16:25 GMT).
2
Wikipedia, Czochralski process, http://en.wikipedia.org/wiki/Czochralski_process (as of Sep. 17, 2007,
16:25 GMT).
3
Wikipedia, Miller index, http://en.wikipedia.org/wiki/Miller_index (as of Sep. 17, 2007, 16:29 GMT).
intercepts by dividing each by the unit cell length along their respective axes, giving us 1, 2, 3 (in
the order x, y, z). We then invert the normalized values to get 1, 1/2, 1/3. To get the smallest
possible whole numbers from this set, we multiply each inverted value by 6, to get 6, 3, 2. At this
point, we can insert parentheses to get the Miller indices for the desired plane: (632).
To properly find and use Miller indices, the following need to be kept in mind. First, if the plane
of interest is parallel to one of your coordinate axes (i.e., if it does not intersect one or two of the
coordinate axes), the intercept values to be found in step (2) above are set to infinity for the
respective axes. For example, the plane designated by the intersects 5, ∞, ∞ (given the unit cell
with sides of length 5 units) is normalized to 1, ∞, ∞, inverted to 1, 0, 0, and turned into the
Miller indices (100).
Second, step (2) intercepts falling in the negative regions of the coordinate system result in a bar
being placed over the relevant index number in the calculated Miller index. An example of such
Miller indices includes the (221)4 plane.
And third, because of symmetry in the unit cells of crystals such as silicon, it is possible to obtain
Miller indices that are effectively equivalent because they are impossible to distinguish from each
other. So, for example, (100), (010), (001), (100), (010), and (001) are equivalent planes. The
appropriate way to refer to this set of identical planes is to use { } braces, and so the above set of
planes would collectively be referred to as the {100} planes.
A procedure analogous to the one above can be used to find the Miller indices for directions (as
opposed to planes):
1. Draw a vector of any length in the direction of interest.
2. Project the vector onto the coordinate axes to find its components in the x, y, and z
directions.
3. Turn the projection values into the smallest possible whole numbers by multiplying them
with a constant.
4. Insert the whole numbers in square brackets [ ].
Similar to Miller indices for planes, an equivalent set of directions is designated using triangular
brackets < >.
In using Miller indices for cubic crystals, it is important to note that a particular plane and the
direction normal to that plane have the same index values, such that, for example, the [101]
direction is normal to the (101) plane. It is also useful to realize that in practice most Miller
indices have zeroes and ones as their index values. High-number indices are rarely encountered.
Reference:
Robert F. Pierret, SEMICONDUCTOR FUNDAMENTALS VOLUME I, MODULAR SERIES ON
SOLID STATE DEVICES 10–13 (2d ed. 1989).
4
Because MS-Word does not allow us to place a bar over the index number 2, this paper designates such
negative indices using underline instead.