# Crystal Planes in Semiconductors

## Miller Indices

All lattice planes and lattice directions are described by a mathematical description known as a Miller Index. This allows the specification, investigation, and discussion of specific planes and directions of a crystal. In the cubic lattice system, the direction [hkl] defines a vector direction normal to surface of a particular plane or facet. The Miller Indices h,k,l are defined as follows:   type: <100> Equivalent directions: ,, type: <110> Equivalent directions: , , , [-1-10], [0-1-1], [-10-1], [-110], [0-11], [-101], [1-10], [01-1], [10-1] type: <111> Equivalent directions: , [-111], [1-11], [11-1]

1. Miller Conventions -- a guide for when to use { } vs ( ) and [ ] vs < >
1. As a crystal is periodic, there exist families of equivalent directions and planes. Notation allows for distinction between a specific direction or plane and families of such.
2. Use the [ ] notation to identify a specific direction (ie [1,0,-1]).
3. Use the < > notation to identify a family of equivalent directions (ie <110>).
4. Use the ( ) notation to identify a specific plane (ie (113)).
5. Use the { } notation to identify a family of equivalent planes (ie {311}).
6. A bar above a index is equivalent to a minus sign.
2. Algorithm for determining [hkl]
1. Miller indices are referenced to the crystallographic axes of a crystal. They therefore do not have to be oriented at right angles, though they correspond to the x, y, z, axes in cubic latice structures. For monoclinic and triclinic crystals, there are four numbers to every Miller index. Cubic latices need only three, however, and an algorithm for determining these Miller indices is given next.
1. (1) Determine the points at which a given crystal plane intersects the three axes, for example at (a,0,0), (0,b,0), (0,0,c). If the plane is parallel an axis, it is said to intersect the axis at infinity.
2. (2) The Miller index for the face is then specified by (1/a,1/b,1/c), where the three numbers are expressed as the smallest integers (common factors are removed). Negative quantities are indicated with an overbar.
2. Examples:
1. A plane intersects the crystallographic axes at (2,0,0), (0,4,0), (0,0,4).
2. Step 1: (1/2,1/4,1/4); multiply by 4 to express as smallest integers.
3. Step 2: (2,1,1) are the Miller indices. This is a (211) plane.
4. The miller indices are (110).
5. Step 1: this plane intersects the crystallographic axes at (1/1,0,0), (0,1/1,0) and (0,0,1/0), or (1,0,0), (0,1,0), and z=infinity (i.e. this plane does not intersect the z axis).

## Cubic Unit Cells

A crystal can always be divided into a fundamental shape with a characteristic shape, volume, and contents. In many crystals, the unit cell may be chosen as a cube, with atoms placed at various points within the cube.

(Silicon and Germanium have the Diamond structure)    Diamond Unit Cell {100} planes {110} planes {111} planes

ANGLE

100

110

010

001

101

## Plane Intersection Angle Calculator

Miller indicies of first plane:
Miller indicies of second plane:

Angle between the planes:

100

0.00

45.0

90.0

90.0

45.0

011

90.0

60.0

45.0

45.0

60.0

111

54.7

35.3

54.7

54.7

35.3

211

35.2

30.0

65.9

65.9

30.0

311

25.2

31.4

72.4

72.4

31.4

511

15.8

35.2

78.9

78.9

35.2

711

11.4

37.6

81.9

81.9

37.6

## Wafer Flats

Purpose and Function

1. Orientation for automatic equipment.

2. Indicate type and orientation of crystal.

Primary flat: The flat of longest length located in the circumference of the wafer. The primary flat has a specific crystal orientation relative to the wafer surface; major flat.

Secondary flat: Indicates the crystal orientation and doping of the wafer. The location of this flat varies.

<100>

Flat at 180 deg for n-type and 90 deg for p-type. <111>

Flat at 45 deg for n-type, no secondary for p-type. very few 6" or 8" <111> wafers are manufactured.

For large crystals no flats are ground. Instead a notch is machined for positioning and orientation purposes.