1. Lecture 2 PH 4891/581, Jan. 9, 2009 This file has parts of Lecture 2. First, here is the link to Dr. Spears’ website on semiconductor physics. Look at the chapter on semiconductor crystal structures, which has some general material on crystals in general. However, the phrase “basis vector” is used incorrectly, illustrating the risks of getting information from the internet, even from well-known persons.link

2. Rotational symmetries 2D lattices:determined by primitive translation vectors a 1 and a 2 Square has 4-fold (90º = 360º / 4) rotation symmetry 90º Rectangle does not: a1a1 a2a2

3. Rotational symmetry of lattices Rotate these in PowerPoint to determine rotational symmetry group Square lattice: Rectangular lattice:

4. Oblique lattice ALL lattices have 180º symmetry!

5. “Systems” (different rotational symmetries) in 2D Oblique (0º, 180º rotation only) Square (0º, 90º, 180º, 270º, 4 mirrors) Rectangular (0º, 180º, 2 mirrors) Hexagonal (0º, 60º, 120º, 180º, 240º, 300º, 6 mirrors)

6. Same symmetries! “Centered rectangular lattice” So there are 5 “Bravais lattices” in 2D. Not done! Rectangular system has 2 possible lattices Rectangular system (symmetries are 0º, 180º rotations & 2 mirrors) Has rectangular lattice: Add atoms at centers of rectangles:

7. So the 4 2D symmetry systems have a total of 5 Bravais lattices Oblique (2-fold rotation only) –One lattice (“oblique”) Square (4-fold rotations, 4 mirrors) –One lattice (“square”) Rectangular 2-fold, 2 mirrors) –TWO lattices: rectangular and – centered rectangular Hexagonal (6-fold rotations, 6 mirrors) -- One lattice (“hexagonal”)