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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.
Rotational symmetries a2 90º a1 2D lattices:determined by primitive translation vectors a1 and a2 Square has 4-fold (90º = 360º / 4) rotation symmetry Rectangle does not:
Rotational symmetry of latticesRotate these in PowerPoint to determine rotational symmetry group Square lattice: Rectangular lattice:
Oblique lattice ALL lattices have 180º symmetry!
“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)
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: • Same symmetries! • “Centered rectangular lattice” • So there are 5 “Bravais lattices” in 2D.
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”)