Miller Index Lecture Notes

Overview

• The Miller index is a method used to designate specific planes in crystal structures.
• It is used to clearly define specific planes in the three-dimensional crystal structure of solids.

Definition and Calculation of the Miller Index

1. Finding Intercepts

   • Find the intercepts of atoms present on the plane with each coordinate axis (X, Y, Z).
   • Express intercepts based on vectors A, B, C: x-axis 3a, y-axis 2b, z-axis 1c

2. Selecting Integers

   • Take integers representing the intercepts: 3, 2, 1

3. Taking Reciprocals

   • Take the reciprocal of each integer: 1/3, 1/2, 1/1

4. Multiplying by the Least Common Multiple

   • Multiply by the least common multiple 6 to eliminate the denominators: 2, 3, 6
   • The result is the Miller index (2, 3, 6)

Necessity of the Miller Index

• Relation with Reciprocal Lattice and Reciprocal Space: Necessary for the theoretical concept of taking reciprocals.
• Solution to Real Space Problems: The reciprocal space transformation increases problem-solving possibilities.

Exercises and Examples

• Example of a Miller Index: Plane 1, 0, 0
  • Exists intercept with the x-axis, no intercepts with y, z axes
  • Integer: 1, ∞, ∞ → Reciprocal: 1, 0, 0

Importance of Crystal Planes

• Impact on Semiconductor Devices: Affects electron movement and semiconductor performance depending on the crystal plane.
• Changes in Chemical Process Conditions: Reaction conditions change according to atomic arrangement.

Practical Application of Miller Index in Silicon Wafers

• In silicon wafers, Miller indices indicate the direction of the crystal plane.
• The diameter of the wafer is expressed in inches (2 inches~12 inches).
• The orientation of the wafer is indicated by the Miller index: e.g., 1, 0, 0 wafer

Physical Properties of Wafers

• Wafers have a single crystal structure and can be cut in a specific direction.
• A diamond cutter can be used to produce a smooth cut surface.

Conclusion

• The Miller index is an essential tool for clearly defining specific planes in the crystal structure of solids.
• It plays a crucial role in the manufacturing of semiconductor devices and processing of silicon wafers.