Crystal Planes in Semiconductors

Miller Indices

type: <100> Equivalent directions: [100],[010],[001]	type: <110> Equivalent directions: [110], [011], [101], [-1-10], [0-1-1], [-10-1], [-110], [0-11], [-101], [1-10], [01-1], [10-1]	type: <111> Equivalent directions: [111], [-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. 
      
   3. Use the [ ] notation to identify a specific direction (ie [1,0,-1]).
      
   4. Use the < > notation to identify a family of equivalent directions (ie <110>).
      
   5. Use the ( ) notation to identify a specific plane (ie (113)).
      
   6. Use the { } notation to identify a family of equivalent planes (ie {311}).
      
   7. 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.
      
   2. (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.
   3. (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.
   4. 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
   

Simple Cubic (sc)
Body-Centered Cubic (bcc)
Face-Centered Cubic (fcc)
Sodium Chloride (NaCl)
Cesium Chloride (CsCl)
ZincBlend (ZnS)
Diamond (C)  (Silicon and Germanium have the Diamond structure)

Diamond Unit Cell	{100} planes	{110} planes	{111} planes

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.

1. These links provide specific information on each orientation.
2. <100>
   1. flat at 180 deg for n-type and 90 deg for p-type.
   2. 
3. <111>
   1. flat at 45 deg for n-type, no secondary for p-type.
   2. 
   3. very few 6” or 8” <111> wafers are manufactured.
4. <110>
   1. do standards exist? email kjc6@et.byu.edu if you know.
5. <311>
   1. do standards exist? email kjc6@et.byu.edu if you know.
6. <511>
   1. do standards exist? email kjc6@et.byu.edu if you know.
7. <711>
   1. do standards exist? email kjc6@et.byu.edu if you know.
8. For large crystals no flats are ground. Instead a notch is machined for positioning and orientation purposes.