semiconductor physics and Devices

Week 2: Crystal Structure (Part 2)

Lecture Topics:

1. Common Crystal Structures

   • Detailed exploration of Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC) structures.

   • Coordination number (number of nearest neighbors) and atomic packing fraction (APF) for each structure.

     • SC: Coordination number = 6, APF = 0.52

     • BCC: Coordination number = 8, APF = 0.68

     • FCC: Coordination number = 12, APF = 0.74

2. Examples of Materials with Different Crystal Structures

   • Simple Cubic (SC): Polonium (Po) as an example of SC structure.

   • Body-Centered Cubic (BCC): Examples include iron (Fe) at room temperature, chromium (Cr), and tungsten (W).

   • Face-Centered Cubic (FCC): Examples include aluminum (Al), copper (Cu), gold (Au), and silver (Ag).

3. Hexagonal Close-Packed (HCP) Structure

   • Description of the HCP structure, which differs from FCC though both have high packing efficiency.

   • Coordination number = 12, APF = 0.74 (same as FCC).

   • Examples of HCP materials: magnesium (Mg), zinc (Zn), titanium (Ti).

4. X-ray Diffraction and Bragg's Law

   • Basics of how X-ray diffraction is used to study crystal structures.

   • Bragg's Law: nλ=2dsin⁡θn \lambda = 2d \sin\thetanλ=2dsinθ

     • nnn: Order of reflection (integer)

     • λ\lambdaλ: Wavelength of X-rays

     • ddd: Interplanar spacing

     • θ\thetaθ: Angle of incidence (Bragg angle)

   • The role of X-ray diffraction in determining crystal structures.

5. Miller Indices

   • Introduction to Miller indices, which are used to describe the orientation of planes in a crystal lattice.

   • Method for determining Miller indices:

     • Determine the intercepts of the plane with the crystal axes.

     • Take the reciprocals of the intercepts.

     • Convert to smallest integer values.

   • Examples of common planes in SC, BCC, FCC structures.

Homework/Exercises:

1. Calculate the atomic packing fraction (APF) for the FCC structure.

2. Identify the Miller indices for a plane that intercepts the xxx-axis at aaa, the y-axis at ∞, and the z-axis at a/2.

3. Use Bragg’s Law to calculate the angle of diffraction for X-rays with a wavelength of 1.54 A˚,  and a crystal with interplanar spacing of 2.0 A˚.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 1: Crystal Structure (continued).

Key Takeaways:

• Crystal structures like SC, BCC, FCC, and HCP are fundamental in understanding material properties.

• X-ray diffraction is a powerful tool for determining crystal structures using Bragg’s Law.

• Miller indices are an essential method for identifying crystallographic planes.

This week expands on the foundational concepts introduced in Week 1, with a focus on specific crystal structures, X-ray diffraction, and crystallographic notation, setting the stage for further study of the properties of solids.