semiconductor physics and Devices

Week 3: Reciprocal Lattice (Part 1)

Lecture Topics:

1. Introduction to the Reciprocal Lattice

   • Concept of the reciprocal lattice as the Fourier transform of the real-space lattice.

   • Importance of the reciprocal lattice in solid state physics, particularly in describing wave vectors, diffraction patterns, and band structure.

2. Mathematical Definition of the Reciprocal Lattice

   • Definition: If a1,a2,a3 are the primitive lattice vectors in real space, the reciprocal lattice vectors b1​,b2​,b3​ are defined as:

The reciprocal lattice vectors are perpendicular to the planes in the crystal and are used to describe diffraction conditions.

1. Physical Interpretation of the Reciprocal Lattice

   • Reciprocal lattice vectors correspond to the periodicity in reciprocal space.

   • The reciprocal lattice provides a framework to describe diffraction conditions, such as those given by Bragg's Law.

   • Introduction to wave vectors in reciprocal space and their relation to crystal symmetry.

2. Reciprocal Lattice of Common Structures

   • SC lattice: The reciprocal lattice of a simple cubic lattice is also simple cubic.

   • BCC lattice: The reciprocal lattice of a body-centered cubic lattice is face-centered cubic (FCC).

   • FCC lattice: The reciprocal lattice of a face-centered cubic lattice is body-centered cubic (BCC).

   • Demonstrating how reciprocal lattices are constructed for different types of crystal structures.

Examples:

• Constructing the reciprocal lattice for a simple cubic lattice.

• Showing that the reciprocal lattice of an FCC structure is a BCC lattice.

• Explaining how reciprocal lattices simplify diffraction calculations.

Homework/Exercises:

1. Derive the reciprocal lattice vectors for a simple cubic crystal structure.

2. Show that the reciprocal lattice of a BCC lattice is an FCC lattice.

3. Calculate the reciprocal lattice vectors for a crystal with lattice constants a=3.0A˚, b=4.0A˚, and c=5.0A˚.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 2: Reciprocal Lattice.

Key Takeaways:

• The reciprocal lattice is a mathematical construct that provides critical insights into wave vectors and diffraction patterns in crystal structures.

• Reciprocal lattice vectors simplify the understanding of X-ray diffraction and electronic band structures.

• Different crystal structures have corresponding reciprocal lattices, which are essential for interpreting diffraction data.

This week introduces the powerful concept of the reciprocal lattice, providing students with the tools they will use to understand diffraction and electronic properties in later chapters.